Math 818 - Table of contents
Chapter 5: Introduction to rings
- Section G: Further interactions between quotients,
products, and isomorphism invariants
- Building integral domains and fields
- Prop 5.126: (1) Let R be a ring with 1 ≠ 0 and let I be an ideal of R.
I = R if and only if I contains a unit.
(2) If R is a field and h: R → S is a ring homomorphism,
then either h is the zero homomorphism or h is injective.
- Def 5.129: An ideal I in a ring R is a proper ideal if I ≠ R.
- Def 5.130: An ideal M in a ring R is a maximal ideal if
M ≠ R and the only ideals of R containing M are M and R.
- Def 5.131: Let R be a commutative ring with 1 ≠ 0. An ideal
P of R is a prime ideal if P ≠ R and whenever a,b ∈ R and
ab ∈ p then either a ∈ P or b ∈ P.
- Thm 5.133: Let R be a commutative ring with 1 ≠ 0, and let I
be an ideal of R.
(a) The ideal I is maximal if and only if R/I is a field.
(b) The ideal I is prime if and only if R/I is an integral domain.
(c) Every maximal ideal of R is prime.
- Examples
- Ideals in Z and Z[x]
- Def 5.135: Let R be a commutative ring with 1 ≠ 0 and let
G be a group. The augmentation ideal in the group ring RG
is the kernel of the augmentation homomorphism h: RG → R
defined by h(∑_{g ∈ G} r_{g}g) =
∑_{g ∈ G} r_{g} for all
∑_{g ∈ G} r_{g}g ∈ RG.
- Ideals in (Z/2Z)C_3 ≅ Z[x]/(2,x^{3}).
- Thm 5.136: (Binomial Theorem):
Let R be a commutative ring with 1 ≠ 0. Then
(r+s)^{n} = ∑_{i=0}^{n} b(n,i)r^{i}s^{n-i}
for all r,s ∈ R and n ≥ 0, where
b(n,i) is the sum 1 + 1 + ··· + 1 of
(n!)/(i!)((n-i)!) copies of 1 in R.
- Def 5.136b: Let R be a commutative ring with 1 ≠ 0 and let I
be an ideal of I. The
radical of the ideal I in R is the
set rad(I) = {r ∈ R | r^{n} ∈ I for some n ≥ 1}.
The ideal I is a radical ideal if rad(I) = I.
The nilradical of the ring R is the ideal rad(0); that is,
set of nilpotent elements of R, denoted N(R).
- The radical of an ideal is an ideal. Every prime ideal of R is radical.
- Thm 5.137: If R is a ring with 1 ≠ 0 and I is a proper ideal
of R, then there is a maximal ideal of R containing I.
- Cor 5.138: If R is a commutative ring with 1 ≠ 0 then
there is a quotient of R that is a field.
- Def 5.140: A partially ordered set, or poset, is
a set A with a relation ≤ on A satisfying the following for all
x,y,z ∈ A: (1) x ≤ x.
(2) If x ≤ y and y ≤ x then x = y.
(3) If x ≤ y and y ≤ z then x ≤ z.
- Axiom 5.141: (Zorn's Lemma) If A is a nonempty poset
satisfying the property that whenever a subset B ⊆ A
is totally ordered (that is,
for all b,b' ∈ B either b ≤ b' or b' ≤ b) then there is
an upper bound for B in A
(that is, an element u_B ∈ A such that b ≤ u_B for all b ∈ B),
then there is a maximal element m ∈ A (that is, there is
an m ∈ A such that whenever x ∈ A and m ≤ x then m = x).
- Def 5.143: Let R be a commutative ring with 1 ≠ 0 and let
S be a multiplicatively closed subset of nonzero divisors of R such that
1 ∈ S and 0 ∉ S. The ring of fractions S^{-1}R
[also known as the localization of R by S]
is defined by: Let X = {r/s | r ∈ R, s ∈ S}, and let ~ be
the equivalence relation on X defined by r/s ~ r'/s' if and only if
rs' = r's. Then S^{-1}R is the set X/~ of equivalence classes,
with addition r/s + r'/s' = (rs'+r's)/(ss') and multiplication
(r/s)(r'/s') = (rr')/(ss') for all r/s,r'/s' ∈ S^{-1}R.
- Prop 5.145: If R is a commutative ring with 1 ≠ 0 and
S is a multiplicatively closed subset of nonzero divisors of R such that
1 ∈ S and 0 ∉ S, then S^{-1}R is a commutative
ring with 1 ≠ 0, and the function i: R → S^{-1}R
defined by i(r) = r/1 for all r ∈ R is an embedding.
- Def 5.147: Let R be an integral domain. The field of fractions
of R is the ring of fractions Frac(R) = (R-{0})^{-1}R.
- Thm 5.148: Let R be an integral domain.
(1) The field of fractions Frac(R)
of R is a field.
(2) R is (isomorphic to) a subring of the field Frac(R).
(3) If F' is any field containing R, then there
is an embedding h: Frac(R) → F' given by h(r/s) = rs^{-1} for
all r ∈ R and s ∈ R-{0}.
- Examples
- Def 5.150: Let R be a commutative ring with 1 ≠ 0,
and let r ∈ R be a nonzerodivisor. The ring R[1/r] is the ring
S^{-1}R where S = {r^{n} | n ≥ 0}.
- Let R be a commutative ring with 1 ≠ 0,
let P be a prime ideal of R, and let S = R - P.
- Thm 5.152: (Homomorphism Building Theorem (HBT) for
Rings of Fractions): Let R and T be commutative rings with 1 ≠ 0, let
S be a multiplicatively closed subset of nonzero divisors of R such that
1 ∈ S and 0 ∉ S, and let j: R → T be a ring homomorphism
satisfying the property that j(s) is a unit of T for all s ∈ S.
Then there is a
unique homomorphism h: S^{-1}R → T such that h(r/1) = j(r)
For all r ∈ R. Moreover, h(r/s) = j(r)j(s)^{-1} for all
r/s ∈ S^{-1}R.
- Thm 5.154: Let R be a commutative ring with 1 ≠ 0, and let
G be a group. Then the group ring RG is a quotient of a ring of
noncommutative polynomials with coefficients in R. Moreover, if G is
abelian, then RG is a quotient of a ring of
commutative polynomials with coefficients in R.
- The Chinese Remainder Theorem
- Def 5.156: Let R be a commutative ring with 1 ≠ 0. Two ideals I,J of R
are comaximal if I + J = R.
- Thm 5.157: (Chinese Remainder Theorem): Let R be a commutative
ring with 1 ≠ 0 and let I_{1},...,I_{n} be ideals of R.
(1) The function h: R → (R/I_{1}) × ··· ×
(R/I_{n}) defined by h(r) = (r+I_{1},...,r+I_{n})
for all r ∈ R is a ring homomorphism with kernel
I_{1} ∩ ··· ∩ I_{n}.
(2) If I_{i} + I_{j} = R for all i ≠ j, then
I_{1} ∩ ··· ∩ I_{n} =
I_{1} ··· I_{n}, h is onto, and
R/(I_{1} ··· I_{n}) ≅
(R/I_{1}) × ··· ×
(R/I_{n}).
- Examples
Chapter 6: "Nice" integral domains and polynomial rings
- Section A: Euclidean domains
- Def 6.1: Let R be a commutative ring and let a,b ∈ R.
(1) The element b is a divisor of a, and a is a multiple of b,
if there is an element x ∈ R with a = bx.
(2) A greatest common divisor, or gcd, of a and b is an element d ∈ R
satisfying d | a, d | b, and whenever [d' | a and d' | b] then d' | d.
- Def 6.2: Let R be a commutative ring with 1 ≠ 0
and let a,b ∈ R with a,b ≠ 0.
A least common multiple, or lcm, of a and b is an element m ∈ R
satisfying a | m, b | m, and whenever [a | m' and b | m'] then m | m'.
- Rmk: If a and b have a lcm m, then the ideal (m)
is the unique largest principal ideal contained in (a) ∩ (b).
- Examples:
- R = Z; every pair of nonzero elements has 2 gcd's.
- R = Z[√(-3)] is an integral domain but a = 4, b = 2 + 2√(-3)
have no gcd.
- Rmk: 3 views on gcd's in Z: Euclidean algorithm (⇝ Euclidean domains),
linear combination d = ra + sb (⇝ PID), product of common prime divisors
(⇝ UFD).
- Def 6.3: A strict partial order < on a set A is an irreflexive (a ≮ a
for all a ∈ A),
asymmetric (a < b implies b ≮ a for all a,b ∈ A), transitive relation on A.
A strict partial order is well-founded if there is no infinite descending
chain a_{0} > a_{1} > a_{2} > ···.
- Examples
- A = N_{0}
- A = any set, f: A → N_{0}, a < b iff f(a) < f(b).
- B = any finite set, A = B^{*}, < is "shortlex order"
- B = {x_{1},...,x_{k}},
A = {x_{1}^{n1}···x_{k}^{nk}
| n_{1},...,n_{k} ∈ N_{0}}, < is "degree lex order"
- Rmk: In order to prove that an algorithm terminates, show that a
well-founded strict partial order decreases with each step of the algorithm.
- Def 6.5: A Euclidean domain, or ED, is an integral domain R together
with a function N: R → N_{0} such that
for all a,b ∈ R with b ≠ 0 there exist elements q,r ∈ R
satisfying a = qb + r with either r = 0 or N(r) < N(b).
- Lemma 6.6: "Euclidean domain" is a ring isomorphism invariant.
- Thm 6.7: Let R be a Euclidean domain. Then for all
a,b ∈ R with b ≠ 0, there is at least one gcd of a and b.
Moreover, if there is an algorithm that upon input of any a',b' ∈ R with b' ≠ 0
outputs a pair q,r ∈ R with a' = qb' + r, then there is an algorithm that
upon input of a,b ∈ R with b ≠ 0 outputs a gcd of a and b.
- Examples:
- R = Z, R = S^{-1}Z
- R = any field
- Prop 6.10: Let R be an integral domain.
(1) R[x] is also an integral domain.
(2) For all p,q ∈ R[x] - {0}, degree(pq) = degree(p) + degree(q).
(3) The units of R[x] are the units of R.
- Thm 6.12: If F is a field, then F[x] is a Euclidean domain.
- Section B: Principal ideal domains
- Def 6.15: A principal ideal domain, or PID, is an
integral domain for which every ideal is principal.
- Lemma 6.16: "PID" is an isomorphism invariant.
- Lemma 6.17: If R is a PID and a,b are nonzero elements of R,
then for any d ∈ R with (d) = (a,b), d is a gcd of a and b.
- Examples/Interactions between isomorphism invariants and constructions
- Thm 6.20: Every Euclidean domain is a PID.
- Z[(1 - √(-19))/2] is a PID that is not a
Euclidean domain.
- Cor 6.22: If F is a field, then F[x] is a PID.
- Lemma 6.23: (Z/2Z)[x,y] is not a PID.
- Lemma 6.26: (1) Z is a PID with quotient Z/6Z
that is not an integral domain.
(2) (Z/2Z)[x] is a PID, and is a quotient of
a non-PID Z[x].
- Def 6.30: Let R be an integral domain.
(1) An element p ∈ R is a prime element if p ≠ 0 and the ideal
(p) is a prime ideal.
(2) An element r ∈ R is irreducible if r is not a unit,
and whenever r = xy with x,y ∈ R then either x or y is a unit.
(3) Two elements r,s ∈ R are associates if there is a
unit u of R such that s = ur.
- Lemma 6.31: Let R be an integral domain and let r,s ∈ R.
(1) The element r is a prime element
if and only if whenever xy=rz for some x,y,z ∈ R then either
x or y is a multiple of p.
(2) If r is irreducible, then r ≠ 0.
(3) The elements r and s are associates if and only if (r) = (s).
- Cor 6.32: If R is a PID and a,b ∈ R, then the gcd of a and b
is unique up to multiplication by a unit.
- Thm 6.34: Let R be an integral domain and let r ∈ R. (1) If r is a prime element,
then r is irreducible. (2) If R is a PID and r is irreducible, then r is a prime element.
- Cor 6.35: If R is a PID, then every nonzero prime ideal of R is maximal.
- Cor 6.36: Let F be a field and let p ∈ F[x].
Then F[x]/(p) is a field if and only if p is irreducible.
- Section C: Unique factorization domains
- Def 6.40: A unique factorization domain, or UFD,
is an integral domain for which every nonzero, non-unit element r ∈ R
can be written as a finite product
r = p_{1} ··· p_{m} with
m ≥ 1 and each p_{i} irreducible, and moreover
whenever r = p_{1} ··· p_{m}
= q_{1} ··· q_{n} with n ≥ 1
and each q_{j} irreducible, then m = n and there is a permutation
σ ∈ S_{m} such that for all i,
p_{i} and q_{σ(i)} are associates.
- Lemma 6.41: "UFD" is an isomorphism invariant.
- Prop 6.43: If R is a UFD and a,b ∈ R, then there is a gcd of a and b in R.
- Lemma 6.44: Let R be a UFD and let r ∈ R. Then r is prime if and only if
r is irreducible.
- Def 6.46: Let R be a commutative ring with 1 ≠ 0.
Let < be the strict partial order on the set A of ideals of R
defined by I < J if and only if I ⊋ J. The ring R
is called Noetherian, or satisfies the ascending chain condition (acc),
if there is no infinite ascending chain
of ideals I_{1} ⊊ I_{2} ⊊ I_{3} ⊊ ...;
that is, if the strict partial order is well-founded.
- Lemma 6.47: Let R be a commutative ring with 1 ≠ 0. Then
R is Noetherian if and only if every ideal of R is finitely generated.
- Lemma 6.49: If R is a Noetherian integral domain, then every nonzero
nonunit element factors into a finite product of irreducible elements.
- Thm 6.51: If R is a Noetherian integral domain satisfying
that every irreducible element is a prime element, then R is a UFD.
- Cor 6.52: Every PID is a UFD.
- Examples
- Cor 6.54: Every Euclidean domain is a UFD.
- Cor 6.55: (Fundamental Theorem of Arithmetic): Z is a UFD.
- Cor 6.56: If F is a field then F[x] is a UFD.
- Rmk 6.58: (Z/2Z)[x,y] is a UFD that is not a PID.
- Rmk 6.60: {fields} ⊊ {ED's} ⊊ {PID's} ⊊ {UFD's} ⊊
{GCD domains} ⊊ {integral domains} ⊊ {commutative rings}
⊊ {rings}
- Section D: Polynomial rings, irreducibles, and UFD's
- Rmk: Common tools for proofs involving a polynomial ring R[x] over
an integral domain R: (1) Induction on polynomial degree. (2) Consider
the ring R[x] as a subring of F[x], where F is the field of fractions of R.
- Rmk: Common tool for proofs involving UFD's: Induction on the number
of irreducible factors in an irreducible factorization.
- Thm 6.65: (Gauss' Lemma): Let R be a UFD with field of
fractions F and let f ∈ R[x]. [Using Thm 5.148(2), view R as a subring of F.]
If f is irreducible in R[x], then f is irreducible in F[x].
- Prop 6.67: Let R be a UFD with field of fractions F, and let f ∈ R[x].
Suppose that the coefficients of f have no common irreducible factor (that is,
1 is a gcd for the coefficients). Then f is irreducible in R[x] if and only if
f is irreducible in F[x].
- Thm 6.69: Let R be a ring. Then R is a UFD if and only if
R[x] is a UFD.
- Cor 6.70: If R is a UFD, then R[x_{1},...,x_{n}] is a
UFD for all n ≥ 1.
- Roots:
- Prop 6.74: Let F be a field and let p ∈ F[x].
Then x-a divides p if and only if p(a) = 0.
- Def 6.75: Let F be a field, p ∈ F[x], and a ∈ F. The
multiplicity of a as a root of p is the greatest integer n
such that (x-a)^{n} divides p.
- Prop 6.77: Let F be a field and p ∈ F[x]. Then p has at
most degree(p) roots, counted with multiplicity.
- Thm 6.78: If G is a finite subgroup of the multiplicative
group of a field, then G is cyclic.
- Def 6.80: Let R be a ring and let p ∈ R[x].
If p = a_{n}x^{n} + ··· + a_{0}
with each a_{i} ∈ R and a_{n} ≠ 0,
then a_{n} is the leading coefficient of p and
a_{0} is the constant term of p.
- Prop 6.81: (Rational root test):
Let R be a UFD, let p ∈ R[x], and let F be the
field of fractions of R. If r/s ∈ F is a root of p in F[x]
and 1 is a gcd of r,s, then r divides the constant coefficient of p
and s divides the leading coefficient of p.
- Checking irreducibility:
- Prop 6.83: Let F be a field and let p ∈ F[x] be a polynomial of degree
2 or 3. Then p has no root in F if and only if p is irreducible in F[x].
- Prop 6.84: Let R be an integral domain and let p ∈ R[x]
be a monic polynomial of positive degree.
If there is a proper ideal I of R such that
the image of p in (R/I)[x] cannot be factored as a product of polynomials
of positive degree, then p is irreducible in R[x].
- Thm 6.85: (Eisenstein's criterion): Let R be an integral
domain and let p ∈ R[x] be a monic polynomial of positive degree.
If there is a prime ideal Q of R containing all of the coefficients
of p except the leading coefficient, and if the constant term of p
is not in Q^{2}, then p is irreducible in R[x].
- Examples
- Rmk 6.88: Strategy/flow chart for proving irreducibility of a nonzero nonunit
polynomial p in F[x] (where F is a field):
(Step a) Let p' = up, where u is the multiplicative inverse of
the leading coefficient of p. (Then p' is monic and degree(p') > 0.)
Find a UFD R such that F = Frac(R) and p' ∈ R[x]
(possibly R = F).
(Step b) Split into cases depending on degree(p'):
(Case b1) Suppose degree(p') ≤ 3. Find a maximal ideal M of R (possibly 0)
such that for all the divisors d of the constant term of p', d is not a root of
p' in (R/M)[x] (and hence the rational root test Prop 6.81 says that p' has no roots
in (R/M)[x]).
(Case b2) Suppose that degree(p') > 3.
Either (Step b2i) find a proper ideal I of R (possibly 0) such that the image of
p' is not a
product of 2 polynomials of positive degree in (R/I)[x], or
(Step b2ii) find a prime ideal Q of R such that the nonleading coefficients
of p' are in Q and the constant term is not in Q^{2}.
- Section E: The word and ideal membership problems, rewriting systems, and Groebner bases
- Rmk: Recall that for any set B, the set
B^{*} is
the set of all words (strings of letters) over B (including the empty word 1).
- Def 6.92: Let G be a group with a finite generating set A.
The word problem (WP) asks whether there is an algorithm that, upon input of
any element ("word") w of (A ∪ A^{-1})^{*}, can determine
whether w = e in G.
- Def 6.93: A rewriting system over A is a subset S ⊆ A^{*} × A^{*}
such that the rewritings of the form:
xuy → xvy whenever (u,v) ∈ S and
x,y ∈ A^{*}
satisfy:
(1) "Termination": There is no infinite chain of rewritings z_{1} →
z_{2} → z_{3} → ···
(2a) Whenever (u'u'',v),(u''u''',w) ∈ S with u',u'',u''',v,w ∈ A^{*}
and u'' ≠ 1, then there is a word z and rewritings vu''' → ··· → z
and u'w → ··· → z.
(2b) Whenever (u'u''u''',v),(u'',w) ∈ S with u',u'',u''',v,w ∈ A^{*}
and u'' ≠ 1, then there is a word z and rewritings v → ··· → z
and u'wu''' → ··· → z.
A rewriting system for a group G is a rewriting system S over
a generating set A' = A ∪ A^{-1} of G
such that {bb^{-1} → 1 | b ∈ A'} ⊆ R and
G ≅ ⟨ A | {u = v | (u,v) ∈ S} ⟩.
- Thm 6.95: If S is a finite rewriting system for a group G over a finite
generating set
A' = A ∪ A^{-1} of G, then every element of G is represented by exactly one
unrewritable word. Moreover, S solves the word problem: A word w ∈
A'^{*} satisfies w = e in G if and only if w rewrites (using S) to 1.
- Examples, termination orderings on B^{*}, and the Knuth-Bendix algorithm
- Def 6.99: Let F be a field, let
R = F[x_{1},...,x_{n}] be a polynomial ring with
variables in the set A = {x_{1},...,x_{n}},
and let I be an ideal in R.
The ideal membership problem (IMP) asks whether there is an algorithm that,
upon input of any element w in R, can determine
whether w ∈ I (that is, whether w + I = 0 + I in R/I).
- Def 6.100: Let F be a field, let A = {x_{1},...,x_{n}}
be a finite set, and let MM(A) = {x_{1}^{a1}
··· x_{n}^{an} |
a_{1},...,a_{n} ∈ N_{0}} be the
set of monic monomials over A.
A Groebner basis in F[x_{1},...,x_{n}] is a subset
S ⊆ MM(A) × F[x_{1},...,x_{n}]
such that the rewritings of the form:
mu + q → mv+q whenever (u,v) ∈ S, m ∈ MM(A), and
q ∈ R[x_{1},...,x_{n}]
satisfy:
(1) "Termination": There is no infinite chain of rewritings z_{1} →
z_{2} → z_{3} → ···
(2) Whenever (ud,v),(u'd,w) ∈ S, d ∈ MM(A) - {1},
and d is a gcd of ud,u'd,
then there is a polynomial p and rewritings
vu' → ··· → p
and uw → ··· → p.
A Groebner basis for an ideal I in F[x_{1},...,x_{n}]
is a Groebner basis S in F[x_{1},...,x_{n}]
such that I = ({u-v | (u,v) ∈ S}).
- Thm 6.101: If S is a finite Groebner basis for an ideal I
in R = F[x_{1},...,x_{n}] where F is a field,
then every element of R/I is represented by exactly one
unrewritable polynomial. Moreover, S solves the ideal membership problem:
A polynomial p ∈ F[x_{1},...,x_{n}]
satisfies p ∈ I (that is, p + I = 0 + I) if and only if p rewrites (using S) to 0.
- Examples, monomial orderings on MM(A), and the Buchberger algorithm
- R = R[x], I = (x^{2}+5)
- R = (Z/2Z)[x], I = (x^{2}+1,x^{3})
- R = (Z/2Z)[x,y], I = (x^{2}y+xy^{2},x^{2}+x+1)
- Thm 6.104: (Hilbert basis theorem): If R is a commutative Noetherian
ring with 1 ≠ 0, then so is R[x_{1},...,x_{n}] for any n > 0.
- Cor 6.106: Let F be a field and R = F[x_{1},...,x_{n}].
If I is an ideal of R, then I has a finite Groebner basis (and hence the
IMP for I in R has a solution).
Chapter 7: Modules
- Section A: Definitions and first examples
- Motivation: Nonrigid actions
- Def 7.1: Let R be a ring with 1 ≠ 0. A left R-module is an abelian group M
(with operation +) together with an action R × M → M of R on M (written
(r,m) → rm) such that for all r,s ∈ R and m,n ∈ M,
(1) (r + s)m = rm + sm, (2) (rs)m = r(sm), (3) r(m + n) = rm + rn,
and (4) 1m = m.
- Def 7.2: For a ring R without 1, a left R-module is an abelian group M
(with operation +) together with an action of R on M satisfying (1-3) of Def 7.1.
- Def 7.3: Let F be a field. A vector space over F is an
F-module.
- Lemma 7.4: Let R be a commutative ring with 1_{R} ≠ 0
and let M be an R-module. Then 0_{R}m = 0_{M} and
(-1_{R})m = -m for all m ∈ M.
- Def 7.5: Let R be a commutative ring with 1_{R} ≠ 0.
An R-algebra is a ring A with 1_{A} ≠ 0
together with a ring homomorphism f: R → A such that
f(1_{R}) = 1_{A} and f(R) is contained in the center of A.
- Examples:
- For a ring R with ideal I: R, 0, I, R/I
- Polynomials, matrices, group rings
- Def 7.5b: Let R be a commutative ring with 1 ≠ 0,
let J be an ideal of R and let N be an R-module.
Then JN = {∑_{k=1}^{n} j_{k}n_{k} |
n ≥ 0, j ∈ J, n ∈ N}.
- JN is a submodule of N.
- Module constructions by change of rings:
- Restricting the action of R to the action of a subring S.
- Changing the action of R to the action of R/I.
- Changing the action of S to the action of R using a homomorphism
: R → S.
- Extending the action of F to an action of F[x] on a vector space V
using a linear transformation : V → V.
- Section B: Homomorphisms
- Def 7.6: Let R be a ring and let M and N be R-modules.
An R-module homomorphism
from M to N is an additive group homomorphsim h: M → N
such that for all r ∈ R and m ∈ M,
h(rm) = rh(m). An R-module homomorphism h is an
R-module isomorphism if h is also a bijection.
- Def 7.7: Let F be a field and let M and N be vector spaces over F.
A linear transformation from M to N is an
F-module homomorphism : M → N.
- Lemma 7.8: Let R be a ring and let M and N be R-modules. A
function h: M → N is an R-module homomorphism if and only if
h(rm+m') = rh(m)+h(m') for all r ∈ R and m,m' ∈ M.
- Examples:
- Lemma 7.10: (1) Let M be a set with
a binary operation + : M → M.
Then M is an abelian group if and only if M is a Z-module.
(2) Let M,N be abelian groups, and h: M → N.
Then h is an abelian group homomorphism if and only if h is a
Z-module homomorphism.
- Def 7.11: Let R be a ring and let M and N be R-modules. Then
Hom_{R}(M,N) denotes the set of all R-module homomorphisms
from M to N,
and End_{R}(M) = End(M) denotes the set Hom_{R}(M,M).
End(M) is called the endomorphism ring of M, and elements of
End(M) are called endomorphisms.
- Prop 7.12: Let R be a commutative ring with 1 ≠ 0
and let M and N be R-modules.
(1) Hom_{R}(M,N) is an R-module with addition and R-action
defined by (f+g)(m) = f(m)+g(m) and (rf)(m) = r(f(m)) for all
f,g ∈ Hom_{R}(M,N), m ∈ M, and r ∈ R.
(2) End_{R}(M) is an R-algebra, with addition and R-action
defined as in (1), and multiplication defined by
(fg)(m) = f(g(m)) for all f,g ∈ Hom_{R}(M,N) and m ∈ M.
- Section C: Constructions: Submodules, product modules, quotient modules
- Submodules:
- Def 7.15: Let R be a ring and let M be an R-module.
An R-submodule of M is a subgroup N ≤ M satisfying
rn ∈ N for all r ∈ R and n ∈ N.
- Lemma 7.16: Let R be a ring with 1 ≠ 0 and let M be an R-module.
(1) A nonempty subset N of M is a submodule of M if and only if
rn+n' ∈ N for all r ∈ R and n,n' ∈ N.
(2) Let N be an R-submodule of M. Then the inclusion map i:N → M is an
R-module homomorphism.
(3) If h: M → M' is an R-module homomorphism, then
the kernel Ker(h) is an R-submodule of M and the image h(M) is
an R-submodule of M'.
- Examples
- Prop 7.17: Let R be a ring and let M be a subset of R.
Then M is an R-submodule of R if and only if M is a left ideal of R.
- Product modules:
- Def 7.19: Let R be a ring.
(a) let M_{1},...,M_{n} be R-modules.
Their direct product module M_{1} ×
··· M_{n} is the abelian group
M_{1} × ··· M_{n}
with ring action given by r(m_{1},...,m_{n}) =
(rm_{1},...,rm_{n})) for all r ∈ R and
m_{i} ∈ M_{i}.
(b) Let M_{α} be an R-module for all α in an
index set J. The direct product of the R-modules
M_{α} is
the Cartesian product Π_{[α ∈ J]} M_{α}
with addition defined by
(m_{α})_{α ∈ J}+(n_{α})_{α ∈ J}
= (m_{α}+n_{α})_{α ∈ J} and
R-action defined by
r(m_{α})_{α ∈ J}
= (rm_{α})_{α ∈ J}.
The direct sum of the R-modules
M_{α} is the R-submodule
⊕_{[α ∈ J]} M_{α}
of the direct product Π_{[α ∈ J]} M_{α}
given by ⊕_{[α ∈ J]} M_{α}
= {(m_{α})_{α ∈ J} |
m_{α} = 0_{α} for all but
finitely many α}.
- Lemma 7.20: Projections from products
of R-module, inclusions into products of R-modules, and
products of R-module homomorphisms are R-module homomorphisms.
- Examples
- Quotient modules:
- Def 7.22: Let R be a ring, let M be an R-module, and let N be
a submodule of M. The quotient module M/N is the quotient group M/N
with R action defined by r(m + N) = rm + N for all r ∈ R and m + N ∈
M/N.
- Lemma 7.23: Let R be a ring, let M be an R-module, and let N be
a submodule of M. The quotient module M/N is an R-module, and the quotient
map q: M → M/N is an R-module homomorphism with kernel Ker(q) = N.
- Thm 7.24: (Module Isomorphism Theorems):
Let R be a ring, and let M be an R-module.
(1ITM): Let N be an R-module and let h: M → N be an
R-module homomorphism. Then Ker(h) is a submodule of M and M/Ker(h)
≅ h(M).
(2ITM): Let A and B be submodules of M, and let
A + B = {a+b | a ∈ A, b ∈ B}. Then A + B is a submodule
of M, A ∩ B is a submodule of A, and (A + B)/B ≅ A/(A ∩ B).
(3ITM): Let A and B be submodules of M with A ⊆ B.
Then (M/A)/(B/A) ≅ M/B.
(LITM):
Let R be a ring, let N be a R-submodule of an R-module M,
and let q: M → M/N be the quotient map.
Then the function Ψ : {R-submodules of M containing N} → {R-submodules
of M/N}
defined by Ψ(K) = q(K) = K/N is a bijection
with inverse defined by Ψ^{-1}(T) = q^{-1}(T)
for each R-submodule T of M/N. Moreover,
Ψ and Ψ^{-1} preserve sums and intersections.
- Examples
- Section D: Generating modules, free modules, and module homomorphism building theorems:
- Generating modules
- Def 7.28: Let R be a ring with 1 ≠ 0 and let M be an R-module.
For a subset A of M, the submodule of M generated by A is
RA = {r_{1}a_{1} + ···
+ r_{n}a_{n} | n ≥ 0 and each
r_{i} ∈ R and a_{i} ∈ A}; if
A = {a} has a single element, the set RA is denoted Ra.
A subset A of M generates M as an R-module
if RA = M.
A module M is finitely generated if there is
a finite subset A of M that generates M. A module M is cyclic
if M = Rm for some m ∈ M.
- Def 7.29: Let F be a field, let V be a vector space over F,
and let A be a subset of F. The span of A is the submodule
FA of V generated by A.
- Lemma 7.30: "Finitely generated" and "cyclic"
are R-module isomorphism invariants.
- Lemma 7.31: Let M be an R-module and let A ⊆ M.
Then RA = ∩_{[A ⊆ N and N is a submodule of M]} N.
- Examples:
- Let R be a ring with 1 ≠ 0. R = R1 is cyclic.
R × R = R(1,0) + R(0,1) is generated by {(1,0),(0,1)}.
R[x] is generated by the set MM({x}) of monic monomials over x,
as an R-module.
- Let M = Z[x,y]. M is generated by {1,x,y} as a ring,
M is generated by MM({y}) as a Z[x]-module, and
M is generated by {kx^{i}y^{j} | k ∈ Z,
i,j ∈ N_{0}} as a group.
- Lemma 7.32: Let R be a ring with 1 ≠ 0,
let M be an R-module, and let N be an R-submodule of M.
(1) If M is finitely generated as an R-module,
then so is M/N.
(2) If N and M/N are finitely generated as R-modules,
then so is M.
- Free modules, bases, and the HBT
- Def 7.33: Let R be a ring and let A be a set.
The free R-module generated by A, denoted F_{R}(A) = F(A),
the set of formal sums F(A) = {r_{1}a_{1} + ···
+ r_{n}a_{n} | n ≥ 0 and each
r_{i} ∈ R and a_{i} ∈ A}
= { ∑_{[a ∈ A]} r_{a}a | each r_{a} ∈ R and
r_{a} = 0 for all but finitely many a},
with addition defined by
(∑_{[a ∈ A]} r_{a}a) +
(∑_{[a ∈ A]} s_{a}a) =
∑_{[a ∈ A]} (r_{a} + s_{a})a,
and R-action defined by
r(∑_{[a ∈ A]} r_{a}a) =
∑_{[a ∈ A]} (rr_{a})a.
The set A is called a (free) basis for F(A).
The free R-module of rank n
is the free module over a set A with |A| = n.
The free F-module of rank n over a field F is also called
affine n-space over F. The free Z-module over a set
A is called a free abelian group, and the free Z-module
of rank n is called the free abelian group of rank n.
- Lemma 7.35: Let R be a ring and let A be a set. The
free module F_{R}(A) is isomorphic (as R-modules) to
the direct sum module
⊕_{a ∈ A} R.
- Notation 7.36: The free R-module of rank n is also written
R^{n} = R × ··· ×R
(with n factors).
- Def 7.41: Let M be an R-module. A linear combination
of finitely many elements m_{1},...,m_{n} of M
is an element of M of the form
r_{1}m_{1} + ··· +
r_{n}m_{n} for some
r_{1},...,r_{n} ∈ R.
- Def 7.44: Let M be an R-module and let A
be a subset of M. The set A is linearly independent if
whenever r_{1},...,r_{n} ∈ R
and a_{1},...,a_{n} are distinct elements of A satisfying
r_{1}a_{1} + ··· +
r_{n}a_{n} = 0, then
r_{1} = ··· =
r_{n} = 0.
Otherwise A is linearly dependent.
- Def 7.45: A subset A of an R-module M is a basis,
or a free generating set, of M, if the set A generates M
and is linearly independent. An R-module M is a free R-module
if M has a basis.
- Prop 7.46: Let M be an R-module, and let A be a basis of M.
(1) If B is a set of the same cardinality as A, then M ≅ F_{R}(B).
(2) If
and r_{1},...,r_{m},r_{1}',...,r_{n}' ∈ R - {0}
and a_{1},...,a_{m},a_{1}',...,a_{n}' ∈ A
satisfy a_{i} ≠ a_{j}
and a_{i}' ≠ a_{j}' for all i ≠ j
and
r_{1}a_{1} + ··· +
r_{m}a_{m} =
r_{1}'a_{1}' + ··· +
r_{n}'a_{n}',
then m = n and there is a permutation σ ∈ S_{n} such that
r_{i} = r_{σ(i)} and
a_{i} = a_{σ(i)} for all i.
- Thm 7.48: (HBT for free R-modules): Let R be a ring,
let M be a free R-module with basis B, let N be an R-module,
and let j: B → N be
any function. Then there is a unique R-module homomorphism
h: M → N such that h(b) = j(b) for all b ∈ B.
- Examples - general
- Def 7.50: For a ring R with 1 ≠ 0,
the basis {e_{1},...,e_{n}} of R^{n},
where e_{i} = (0,...,0,1,0,...,0) (with 1 in the i-th coordinate)
is the standard basis of R^{n}.
- The empty set is linearly independent.
- Def 7.52: Let N be an R-module.
The annihilator of N in R is
ann_{R}(N) = {r ∈ R | rn = 0 for all n ∈ N}.
- Lemma 7.53: Let M be an R-module.
(1) ann_{R}(M) is an ideal of R.
(2) For any m ∈ M, the singleton set {m}
is linearly independent if and only if ann_{R}(Rm) = 0.
- For a commutative ring R with 1 ≠ 0, R[x] is a free R-module
with basis MM({x}) = set of monic monomials over x.
- For a commutative ring R with 1 ≠ 0 and group G, the group
ring RG is a free R-module with basis G.
- Prop 7.55: Let R be a commutative ring with 1 ≠ 0
and let I be an ideal of R. If I is not principal, then I is not
a free R-module.
- Example - vector spaces
- Lemma 7.58:
Let V be a vector space over a field F. If S is
a linearly independent subset of V and v ∈ V is not in the span
of S, then S ∪ {v} is also linearly independent.
- Prop 7.59: Let V be a vector space over a field F, and suppose
S ⊆ T ⊆ V satisfy that S is linearly independent and
Span(T) = V. Then there is a basis B of V satisfying
S ⊆ B ⊆ T.
- Rmk: Prop 7.59 says that every linearly independent set extends
to a basis, and every spanning set restricts to a basis.
- Thm 7.60: Every vector space over a field F is a
free F-module.
- R is a free Q-module.
- Bases for R^{2} and R^{3}
- Example - abelian groups
- Z/2Z is a Z-module that is not free.
- Def 7.62: A group G with is torsion-free if
for each g ≠ e there is no n_{g} > 0 such that
g^{ng} = e.
- Thm 7.63: Let G be a finitely generated abelian group.
Then G is a free abelian group if and only if G is torsion-free.
- Rmk/Example: The forward direction of Thm 7.63 holds when G is not
finitely generated, but G = Q shows that the converse
direction fails.
Chapter 8: Linear algebra, module classification theorems, and canonical forms
- Section A: Classification of vector spaces, and dimension
- Lemma 8.1: ("Replacement Property"): Let B be a (finite) basis of a
a vector space V and let C = {c_{1},...,c_{n}} be a
linearly independent subset of V. Then there exist distinct
elements b_{1},...,b_{n} ∈ B such that
(B - {b_{1},...,b_{n}}) ∪ {c_{1},...,c_{n}}
is also a basis of V.
- Prop 8.3: Let V be a vector space over F and let B,B' be
bases of V. Then B and B' have the same cardinality.
- Thm 8.5: (Classification of finitely generated vector spaces):
Let F be a field. (1) Every finitely generated vector space over F
is isomorphic to F^{m} for some m ≥ 0.
(2) For any m,n ∈ N_{0},
F^{m} ≅ F^{n} if and only if m = n.
- Cor 8.6: Let R be a commutative ring with 1 ≠ 0
and let m,n ∈ N. Then
R^{m} ≅ R^{n} if and only if m = n.
- Rmk: Cor 8.6 is a classification of finitely generated
free R-modules.
- Def 8.8: Let V be a vector space over a field F. The dimension
dim(V) of V is the cardinality of any basis of V.
- Cor 8.9: Two vector spaces V,V' over a field F are isomorphic
if and only if dim(V) = dim(V').
- Rmk: Cor 8.6 and 8.8 can also
can be recast as "Dimension of free modules is
a module isomorphism invariant".
- Thm 8.10: Let W be a subspace of a vector space V. Then
dim(V) = dim(W) + dim(V/W).
- Cor 8.12: Let f: V → W be a linear transformation.
Then dim(Ker(f)) + dim(f(V)) = dim(V).
- Cor 8.13: Let V and W be finite dimensional vector spaces
with dim(V) = dim(W), and let f: V → W be a linear transformation.
TFAE: (1) f is an isomorphism. (2) f is injective. (2') Ker(f) = 0.
(3) f is surjective. (3') f(V) = W. (4) f maps a basis of V
to a basis of W.
- Def 8.15: Let f: V → W be a linear transformation.
The nullspace of f is Ker(f). The rank of f is dim(f(V)).
The map f is nonsingular if Ker(f) = 0.
- Prop 8.17: Let R be a ring with 1 ≠ 0, let
M and N be R-submodules of an R-module L
(1) The
following are equivalent:
(1a) M ∩ N = {0}.
(1b) The function f: M ⊕ N → M + N defined by
f(m,n) = m + n is an R-module isomorphism.
(1c) For all p ∈ M + N there are
unique m ∈ M and n ∈ N such that p = m + n.
If (1a-c) hold, we say that M + N is the internal direct product of M and N
and write M + N = M ⊕ N.
(2) Let V and W be finite dimensional vector subspaces
of a vector space U over a field F. The following are equivalent:
(2a) V ∩ W = {0}. (2b) dim(V ⊕ W) = dim(V) + dim(W).
- Section B: Matrices and linear algebra
- Linear transformations, ordered bases, and matrices
- Def 8.20: Let F be a field, let V be an F-vector space of
dimension n with ordered basis B = {b_{1},...,b_{n}},
let W be an F-vector space of dimension m with ordered basis
C = {c_{1},...,c_{m}},
and let f: V → W be a linear transformation.
Then [f]_{B}^{C} is the m × n matrix
with entry t_{ij} ∈ F in row i column j determined by
the equation f(b_{j}) =
∑_{i=1}^{m} t_{ij}c_{i}.
- Example
- Def 8.21: Let R be a commutative ring with 1 ≠ 0 and let
f: R^{n} → R^{m} be an R-module homomorphism.
For each 1 ≤ j ≤ n let e_{j} = (0,...,0,1,0,...,0) ∈ R^{n},
with the nonzero entry in the j-th coordinate, and for each
1 ≤ i ≤ m let e_{i}' = (0,...,0,1,0,...,0) ∈ R^{m},
with the nonzero entry in the i-th coordinate.
Then [f]=[f]_{{ej}}^{{ei'}}
is the m × n matrix
with entry t_{ij} ∈ F in row i column j (uniquely) determined by
the equation f(b_{j}) =
∑_{i=1}^{m} t_{ij}c_{i}.
- Def 8.23: Let R be a commutative ring with 1 ≠ 0. Let M_{m,n}(R)
denote the set of matrices with m rows, n columns, and entries in R.
Define addition of matrices by addition of corresponding entries, and
define left multiplication by r ∈ R on a matrix M by left
multiplication by r on each of the entries of M.
- Prop 8.24: Let F be a field, let V be an F-vector space of
dimension n with ordered basis B = {b_{1},...,b_{n}},
let W be an F-vector space of dimension m with ordered basis
C = {c_{1},...,c_{m}},
and let f: V → W be a linear transformation.
Then the function
h=h_{B,C}: Hom_{F}(V,W) → Mat_{m,n}(F)
defined by h(f) = [f]_{B}^{C}
is an isomorphism of F-vector spaces.
- Examples
- Lemma 8.25: If V,W,X are finite dimensional vector spaces with ordered
bases B,C,D and if f: V → W and g: W → X are linear transformations,
then [g ∘ f]_{B}^{D} =
[g]_{C}^{D}[f]_{B}^{C} (where the latter product
is the usual matrix multiplication).
- Def 8.27: Let F be a field. A matrix P ∈ M_{n,n}(F) is
invertible if there is a matrix Q ∈ M_{n,n}(F)
satisfying PQ = I_{n} = QP. The general linear group
GL_{n}(F) is the group of invertible n × n matrices over F, and
for an F-vector space V, the general linear group
GL(V) is the group of invertible
linear transformations : V → V.
- Cor 8.28: Let F be a field, let V be a vector space of dimension n,
and let B be an ordered basis of V.
The function h=h_{B,B} : Hom_{F}(V,V) → Mat_{n,n}(F),
defined by h(f) = [f]_{B}^{B} for all f ∈ Hom_{F}(V,V),
is an isomorphism of F-algebras.
Hence the restriction of h to the units in the
domain and codomain is a group isomorphism h| : GL(V) → GL_{n}(F).
- Changing bases
- Def 8.30: Let R be a commutative ring with 1 ≠ 0, let M be a free R-module
of finite rank n, and let B = {b_{1},...,bn} be a basis for M.
An elementary basis change operation on the basis B
is one of the following three types of operations:
(1) Replacing b_{i} by b_{i} + rb_{j} for some
i ≠ j and some r ∈ R.
(2) Replacing b_{i} by ub_{i} for some i and some unit u of R.
(3) Swapping the indices of b_{i} and b_{j} for some i ≠ j.
- Def 8.31: Let R be a commutative ring with 1 ≠ 0.
An elementary row operation on a matrix A ∈ Mat_{m,n}(R)
is one of the following three types of operations:
(1) Adding (an element of R times a row of A) to a different row of A.
(2) Multiplying a row of A by a unit of R.
(3) Interchanging two rows of A.
- Def 8.32: Let R be a commutative ring with 1 ≠ 0.
An elementary matrix over R is an n × n matrix obtained from I_{n}
by applying a single elementary row operation.
(1) For r ∈ R and 1 ≤ i,j ≤ n
with i ≠ j, let
E_{ij}(r) be the
matrix with 1 in the (i,i) position for all i, r in the (i,j) position,
and 0 in the (k,l) position for all other (k,l).
(2) For u ∈ R^{x} and
1 ≤ i ≤ n let E_{i}(u) denote the matrix with (i,i) entry u,
(j,j) entry 1 for all j ≠ i, and (k,l) entry 0 for all k ≠ l.
(3) For 1 ≤ i,j ≤ n
with i ≠ j, let E_{(i j)} denote the matrix with
1 in the (i,j) and (j,i) positions and in the (l,l) positions for all l ∉ {i,j},
and 0 in all other entries.
- Prop 8.34: Let R be a commutative ring with 1 ≠ 0, and let
E be an n × n elementary matrix.
If A ∈ Mat_{n,q}(R), then the product matrix EA is the result of
performing the corresponding elementary row operation on A.
If B ∈ Mat_{m,n}(R), then the product matrix BE is the result of
performing the corresponding elementary column operation on A.
- Thm 8.36: Let F be a field and let
A be an n × n matrix over F. Then A is invertible if and only if
A is a product of elementary matrices.
- Cor 8.37: Let F be a field and let V be a finite dimensional vector
space over F. Let B and B' be linearly independent subsets
of V.
Then B and B' are bases for the same subspace of V if and only if
there is
a finite sequence of elementary basis change operations from B to B'.
- Def 8.39: Let F be a field.
(1) Let V be an F-vector space. Two linear transformations f,g: V → V
are similar if there is a linear transformation h: V → V
such that g = hfh^{-1}.
(2) Two n × n matrices A and B with entries in R are similar
if there is an invertible n × n matrix P such that B = PAP^{-1}.
- Cor 8.40: Let V be a finite dimensional vector space over a field F,
and let B and B' be bases of V.
(1) Let Id_{V} be the identity map : V → V. Then
[Id_{V}]_{B}^{B'} is an invertible matrix
(called a change of basis matrix).
(2) Let W be a finite dimensional vector space with bases C and C',
and let f: V → W be a linear transformation. Then
[f]_{B'}^{C'} = [Id_{W}]_{C}^{C'}
[f]_{B}^{C} [Id_{V}]_{B'}^{B}.
(3) Let g : V → V be a linear transformation. Then
[g]_{B}^{B} and [g]_{B'}^{B'} are similar.
- Def 8.41: Let F be a field.
The column rank of a matrix M ∈ Mat_{m,n}(F)
is the maximum number of linearly independent columns of M (viewed as
elements of F^{m}).
- Prop 8.42: Let F be a field,
let f: F^{n} → F^{m}, and let
B = {b_{1},...,bn}
and C = {c_{1},...,cm}
be the standard ordered bases of F^{n} and F^{m}, respectively.
Then rank(f) equals the column rank of the matrix
[f]_{B}^{C}.
- Def 8.43: Let F be a field and let A ∈ Mat_{m,n}. The
row reduced echelon form rref of A is a matrix A'
obtained from A (using
"Gaussian elimination") by elementary row operations satisfying the properties that
for each 1 ≤ i ≤ m
the first nonzero entry a_{iji} of the i-th row
of A' is 1, all other entries in the j_{i}-th column are 0, and
j_{1} < j_{2} < ··· j_{r},
where r is the number of nonzero rows of A'. Then numbers j_{i} are
the pivots of A'.
- Prop 8.44: Let F be a field,
let f: F^{n} → F^{m}, and let
B = {b_{1},...,bn}
and C = {c_{1},...,cm}
be the standard ordered bases of F^{n} and F^{m}, respectively.
If A' is an rref of [f]_{B}^{C} with pivots
j_{1},...,j_{r}, then
{f(b_{j1}),...,f(b_{jr})}
is a basis for f(F^{n}).
- Examples
- Determinants
- Def 8.45: Let R be a commutative ring with 1 ≠ 0. For any
v_{1},...,v_{n} ∈ R^{n},
let [v_{1} | ··· | v_{n}]
denote the n × n matrix with the v_{i}'s as columns.
- Def 8.46: Let R be a commutative ring with 1 ≠ 0.
A determinant function is a function d : Mat_{n,n}(R) → R
for some n ≥ 1 satisfying the following:
(1) d is multilinear
on columns; that is, whenever 1 ≤ i ≤ n,
v_{1},...,v_{n},v_{i}' ∈ R^{n},
and r ∈ R, then
d([v_{1} | ··· | v_{i-1} |
v_{i}+v_{i}' | v_{i+1} |
··· | v_{n}]) =
d([v_{1} | ··· | v_{n}]) +
d([v_{1} | ··· | v_{i-1} |
v_{i}' | v_{i+1} |
··· | v_{n}])
and
d([v_{1} | ··· | v_{i-1} |
rv_{i} | v_{i+1} |
··· | v_{n}]) =
rd([v_{1} | ··· | v_{n}])
(2) d is alternating; that is,
whenever v_{i} = v_{j} for some i ≠ j, then
d([v_{1} | ··· | v_{n}]) = 0.
(3) d(I_{n}) = 1.
- Prop 8.48: Let R be an integral domain and
let d : Mat_{n,n}(R) → R be a determinant function, and
let M ∈ Mat_{n,n}(R).
(1) For any r ∈ R and 1 ≤ i,j ≤ n with i ≠ j,
d(M E_{ij}(r)) = d(M).
(2) For any u ∈ R^{x} and 1 ≤ i ≤ n,
d(M E_{i}(u))
= u d(M).
(3) For any 1 ≤ i,j ≤ n with i ≠ j,
d(M E_{(i j)}) = -d(M).
(4) If v_{1},...,v_{n} ∈ R^{n}
are linearly dependent, then
d([v_{1} | ··· | v_{n}]) = 0.
- Thm 8.50: For any commutative ring R with 1 ≠ 0 and integer n ≥ 1,
there is a unique determinant function; moreover, this function
det : Mat_{n,n}(R) → R is given by
det([a_{ij}]) = ∑_{[σ ∈ Sn]}
sign(σ) ∏_{i=1}^{n} a_{iσ(i)}.
- Rmk: The function
det is a (homogeneous) polynomial in the a_{ij}'s of degree n,
with n! terms.
- Prop 8.52: Let R be a commutative ring with 1 ≠ 0.
If A,B ∈ Mat_{n,n}(R) then det(AB) = det(A) det(B).
- Thm 8.53: (1) Let F be a field and let A ∈ Mat_{n,n}(F).
Then A is invertible if and only if det(A) ≠ 0.
(2) Let R be a commutative ring with 1 ≠ 0 and let A ∈ Mat_{n,n}(R).
Then A is invertible if and only if det(A) ∈ R^{x}.
- Def 8.54: Let V be a finite dimensional vector space. The
determinant det(f) of a linear transformation f: V → V
is defined to be det([f]_{B}^{B}) for any basis B of V.
- Lemma 8.55: The determinant of a linear transformation is well-defined.
- Def 8.57: Let V be a vector space over a field F.
Let t: V → V be a linear transformation. A nonzero
element v ∈ V satisfying t(v) = λ v for some λ ∈ F
is an eigenvector of t with eigenvalue λ.
Let A ∈ Mat_{n,n}(F).
A nonzero
element v ∈ F^{n} satisfying Av = λ v for some λ ∈ F
is an eigenvector of A with eigenvalue λ.
- Prop 8.58: Let V be a vector space over a field F and
let t: V → V be a linear transformation.
(1) If λ is an eigenvalue of t then
{eigenvectors of t with eigenvalue λ} ∪ {0}
is a subspace of V (called the eigenspace of the eigenvalue λ).
(2) If λ_{1},...,λ_{k}
are distinct eigenvalues of t, and for each 1 ≤ i ≤ k,
v_{i} is an eigenvector of t with eigenvalue λ_{i}, then
{v_{1},...,v_{k}} is linearly independent.
- Section C: Classification of finitely generated modules over a PID
- Presenting modules
- Def 8.60: Let R be a commutative ring with 1 ≠ 0,
let A ∈ Mat_{m,n}(R), and let t_{A}: R^{n} →
R^{m} be the R-module homomorphism represented by A
over the standard bases. The R-module presented by A
is the R-module R^{m}/t_{A}(R^{n}).
- Prop 8.62: Let R be a Noetherian integral domain.
If M is a finitely generated
module, then every submodule of M is also finitely generated.
- Prop 8.64: Let be a Noetherian integral domain
and let M be a finitely generated R-module. Then there is a
matrix A that presents M.
- Def 8.65: A sequence of functions
f_{1}: X_{1} → X_{2},
f_{2} : X_{2} → X_{3},...
is exact if for each index i, f_{i}(X_{i}) =
Ker(f_{i+1}).
- The classification theorem and Smith Normal Form
- Example
- Prop 8.67: Let R be a commutative ring with 1 ≠ 0 and
let A ∈ Mat_{m,n}(R) and B ∈ Mat_{p,q}(R)
for some m,n,p,q ≥ 1. Then A and B present isomorphic R-modules if and only if
B can be obtained from A by any finite sequence of operations
of the following form: (1) An elementary row operation.
(2) An elementary column operation.
(3) Deletion of the j-th column and i-th row of A if
Ae_{j} = e_{i}
(where e_{i} denote the i-th
standard basis vector in any R^{k}).
(4) The reverse of operation (3) (insertion of a row and column satisfying the
restriction).
(5) Deletion of a column of all 0's.
(6) The reverse of (5) (insertion of a column of all 0's).
- Thm 8.70: (Smith Normal Form (SNF)): Let R be a PID and let
A ∈ Mat_{m,n}(R). Then there is a sequence of elementary
row and column operations that transform A into a matrix A' = [a_{ij}'] such
that all non-diagonal entries of A' are 0, and the diagonal entries
of A' satisfy a_{11}' | a_{22}' | a_{33}' | ···.
Moreover, the number j of nonzero entries of A' is uniquely determined by A,
and the nonzero diagonal entries a_{11}',...,a_{jj}' are
unique up to associates.
- Lemma 8.72: Let R be a commutative ring with 1 ≠ 0, let m ≥ n,
let A = [a_{ij}] ∈ Mat_{m,n}(R) be a matrix such
that all non-diagonal entries of A are 0, and let M be the R-module presented by A.
Then M ≅ R/(a_{11}) ⊕ ··· ⊕
R/(a_{nn}) ⊕ R^{m-n}.
- Thm 8.75: (Classification of Finitely Generated Modules
over a PID (CFGMPID)): Let R be a PID and let M be a finitely
generated module. Then there exist r ≥ 0, k ≥ 0, and
nonzero nonunit elements d_{1},...,d_{k} of R
satisfying d_{i} | d_{i+1} for all i, such that
M ≅ R^{r} ⊕ R/(d_{1}) ⊕ ···
⊕ R/(d_{k}). Moreover r and k are uniquely determined by M,
and the d_{i} are unique up to associates.
- Def 8.76: Let R be a PID, let r ≥ 0, k ≥ 0, and
let d_{1},...,d_{k} be nonzero nonunit elements of R
satisfying d_{i} | d_{i+1} for all i.
Let M be the R-module
M ≅ R^{r} ⊕ R/(d_{1}) ⊕ ···
⊕ R/(d_{k}).
The free rank of M is the integer r. The elements
d_{1},...,d_{k} are the invariant factors of M.
- Cor 8.77: (Classification of Finitely Generated Modules
over a PID using Elementary Divisors (CFGMPIDED)):
Let R be a PID and let M be a finitely
generated module. Then there exist r ≥ 0, s ≥ 0, and
prime elements p_{1},...,p_{s} of R,
and e_{1},...,e_{s} ≥ 1, such that
M ≅ R^{r} ⊕ R/(p_{1}^{e1})
⊕ ···
⊕ R/(p_{s}^{es}).
Moreover r and k are uniquely determined by M,
and the list p_{1}^{e1},...,
p_{s}^{es} is unique up to associates and reordering.
- Def 8.78: Let R be a PID, let r ≥ 0, s ≥ 0,
let p_{1},...,p_{s} be prime elements of R,
and let e_{1},...,e_{s} ≥ 1.
Let M be the R-module
M ≅ R^{r} ⊕ R/(p_{1}^{e1})
⊕ ···
⊕ R/(p_{s}^{es}).
The elements p_{1}^{e1},...,
p_{s}^{es} of R are the elementary divisors of M.
- Example - abelian groups
- Rmk 8.80: The Classification of Finitely Generated Abelian Groups
CFGAG, TOC Thm 4.88) and its elementary divisor version (TOC Thm 4.85)
are the CFGMPID and CFGMPIDED, respectively, applied to the ring R = Z.
- Section D: Canonical forms for endomorphisms
- Rational Canonical Form - Existence/uniqueness
- Def 8.83: Let F be a field, let V be a finite dimensional
vector space over F, and let t: V → V be a linear transformation.
The F[x]-module V_{t} is defined to be
the vector space V with the
unique F[x]-action satisfying xv = t(v) for all v ∈ V.
[That is, (r_{n}x^{n} + ··· +
r_{0})v = r_{n}t^{n}(v) + ··· +
r_{0}v for all
r_{n}x^{n} + ··· +
r_{0} ∈ F[x].)
- Prop 8.84: Let F be a field, let V be a finite dimensional
vector space over F, and let t: V → V be a linear transformation.
Then there are monic polynomials g_{1},...,g_{k} ∈ F[x]
of positive degree such that g_{i} | g_{i+1}
for all i and there is an F[x]-module isomorphism
V_{t} ≅ F[x]/(g_{1}) ⊕ ···
⊕ F[x]/(g_{k}). Moreover, k and the g_{i} are
uniquely determined by V.
- Def 8.85: The polynomials g_{1},...,g_{k}
in Prop 8.84 are the invariant factors of the linear transformation t.
- Def 8.87: Let F be a field and let
p = x^{n} + r_{n-1}x^{n-1} + ··· +
r_{0} ∈ F[x] be a monic polynomial. The companion matrix of p,
denoted C(p), is the matrix [c_{ij}] in M_{n,n}(F)
satisfying c_{i,i-1} = 1 for all 2 ≤ i ≤ n,
c_{in} = -r_{i-1} for all 1 ≤ i ≤ n,
and c_{ij} = 0 for all other i,j.
- Def 8.88: Let F be a field and let A ∈ Mat_{m,m}(F)
and B ∈ Mat_{n,n}(F). The matrix A ⊕ B
is the matrix [c_{ij}] in M_{m+n,m+n}(F) defined by
c_{ij} = a_{ij} for all 1 ≤ i,j ≤ m,
c_{ij} = b_{i-m,j-m} for all m+1 ≤ i,j ≤ m+n,
and c_{ij} = 0 otherwise.
- Thm 8.90: Let F be a field, let V be a finite dimensional vector space, and let
t: V → V be a linear transformation. Then there is an ordered
basis B for V such that
[t]_{B}^{B} = C(g_{1}) ⊕ ···
⊕ C(g_{k}) for some
monic polynomials g_{1},...,g_{k} ∈ F[x]
of positive degree satisfying g_{i} | g_{i+1}
for all i. Moreover, the g_{i} are the invariant factors for t.
- Def 8.91: Let F be a field.
(1) Let V be a finite dimensional vector space, and let
t: V → V be a linear transformation with invariant factors
g_{1},...,g_{k} (with g_{i} | g_{i+1}
for all i).
The matrix C(g_{1}) ⊕ ···
⊕ C(g_{k}) is the rational canonical form of t.
(2) Let A ∈ Mat_{n,n}(F), and let t: F^{n} → F^{n}
be the linear transformation such that A = [t]_{E}^{E},
where E is the standard basis of F^{n}. The
rational canonical form of A is the rational canonical form of t_{A}.
- Rational canonical form - computation
- Thm 8.96: Let F be a field, let V be an F-vector space of dimension n, let
t: V → V be a linear transformation,
let B be an ordered basis for V, and let A = [t]_{B}^{B}.
Then the matrix xI_{n} - A ∈ Mat_{n,n}(F[x])
presents the F[x]-module V_{t}.
- Cor 8.97: Let F be a field, let V be an F-vector space of dimension n, let
t: V → V be a linear transformation, let B be an ordered basis of V,
and let A = [t]_{B}^{B}. The invariant factors of t
are the nonzero nonunit entries in the Smith normal form of the matrix
xI_{n} - A.
- Def 8.99: Let F be a field. (1) Let A ∈ Mat_{n,n}(F).
The characteristic polynomial of A,
denoted c_{A}, is the polynomial
c_{A} = det(xI_{n} - A).
(2) Let V be an F-vector space of dimension n,
and let
t: V → V be a linear transformation. The
characteristic polynomial of t, denoted c_{t},
is the polynomial c_{A} for a matrix
A = [t]_{B}^{B} with respect to some ordered basis B of V.
- Lemma 8.100: Let F be a field. (1) Let V be an F-vector space of dimension n,
and let
t: V → V be a linear transformation. The characteristic
polynomial c_{t} of t is well-defined.
(2) For any A ∈ Mat_{m,m}(F)
and B ∈ Mat_{n,n}(F),
c_{A ⊕ B} = c_{A}c_{B}.
- Def 8.102: Let F be a field. (1) Let A ∈ Mat_{n,n}(F).
A minimal polynomial of A,
denoted m_{A}, is a monic polynomial of least degree such that
m_{A}(A) = 0.
(2) Let V be an F-vector space of dimension n,
and let
t: V → V be a linear transformation. The
minimal polynomial of t, denoted m_{t},
is the monic polynomial generating the annihilator ideal
ann_{F[x]}(V_{t}) in the PID F[x].
- Lemma 8.103: Let F be a field. (1) Let V be an F-vector space of dimension n,
and let
t: V → V be a linear transformation. The minimal
polynomial m_{A} of A = [t]_{B}^{B}
satisfies m_{A} = m_{t}, and hence
is independent of the choice of ordered basis B of V.
(2) For any polynomial p ∈ F[x], m_{C(p)} = c_{C(p)} = p.
- Thm 8.105: Let F be a field, let V be a
finite dimensional F-vector space,
and let
t: V → V be a linear transformation with invariant factors
g_{1},...,g_{k} (with g_{i} | g_{i+1}
for all i).
(1) c_{t} = g_{1} ··· g_{k}.
(2) m_{t} = g_{k}.
(3) (Cayley-Hamilton Thm): m_{t} | c_{t}, and hence
c_{t}(t) = 0.
(4) c_{t} | m_{t}^{k}.
(5) Let f ∈ F. The following are equivalent:
(5a) f is an eigenvalue of t. (5b) f is
a root of m_{t}. (5c) f is a root of c_{t}.
- Thm 8.107: Let F be a field and let
A,A' ∈ Mat_{n,n}(F). The following are equivalent.
(a) A and A' are similar matrices.
(b) A and A' have the same
rational canonical form.
(c) A and A' have the same invariant factors.
(d) xI_{n} - A and xI_{n} - A' have the same
Smith normal form.
(e) For every field K such that F is a subfield of K,
A and A' are similar matrices
in Mat_{n,n}(K).
- Jordan Canonical Form
- Def 8.110: Let F be a field, let n > 0, and let r ∈ F.
The Jordan block J_{n}(r) is the n × n matrix
over F with entries a_{ij} (in row i column j) satisfying
a_{ii} = r for all 1 ≤ i ≤ n,
a_{i,i+1} = 1 for all 1 ≤ i ≤ n-1, and
a_{ij} = 0 for all other i,j.
- Thm 8.112: Let F be a field, let V be a finite dimensional vector space, and let
t: V → V be a linear transformation satisfying the property that
the characteristic polynomial c_{t} of t factors completely into linear terms.
Then there is an ordered
basis B for V such that
[t]_{B}^{B} = J_{e1}(r_{1})
⊕ ···
⊕J_{es}(r_{s})
such that each r_{i} ∈ F is a root of c_{t}
and each e_{i} ≥ 1.
Moreover, the polynomials
(x - r_{1})^{e1},...,(x - r_{s})^{es}
are the elementary divisors of the F[x]-module V_{t},
and this expression for [t]_{B}^{B}
is unique up to ordering of the Jordan blocks.
- Def 8.114: Let F be a field.
(1) Let V be a finite dimensional vector space, and let
t: V → V be a linear transformation with elementary divisors
(x - r_{1})^{e1},...,(x - r_{s})^{es}.
The matrix J_{e1}(r_{1})
⊕ ···
⊕J_{es}(r_{s}) is a Jordan canonical form of t.
(2) Let A ∈ Mat_{n,n}(F), and let t: F^{n} → F^{n}
be the linear transformation such that A = [t]_{E}^{E},
where E is the standard basis of F^{n}. A
Jordan canonical form of A is a Jordan canonical form of t_{A}.
- Def 8.116: Let F be a field.
(1) Let V be a finite dimensional
vector space, and let
t: V → V be a linear transformation. Then t is diagonalizable
if there is a basis B for V such that the matrix
[t]_{B}^{B} is a diagonal matrix.
(2) Let A ∈ Mat_{n,n}(F). Then A is diagonalizable
if A is similar to a diagonal matrix.
- Thm 8.117: Let F be a field, let V be a finite
dimensional vector space, and let t: V → V be a linear transformation.
The following are equivalent:
(a) t is diagonalizable.
(b) t has a Jordan canonical form A and A is a diagonal matrix.
(c) t has a Jordan canonical form and the elementary divisors are
all of the form (x - r)^{1} (with r ∈ F).
(d) Each invariant factor of t is a product of linear polynomials with
no repeated linear factors.
(e) The minimum polynomial of t is a product of linear polynomials with
no repeated linear factors.
Chapter 9: Field theory
- Section A: Field extensions
- Definitions and first examples
- Motivation: Categories, functors, and connections between fields and groups
- Def 9.1: An extension field of a field F is a field K containing F.
- Def 9.2: An extension field K of a field F is denoted K/F.
- Prop 9.4: Let F be a field. (1) The characteristic char(F) is either 0 or
a prime number p. (2) Exactly one of the fields Q or Z/pZ
for some prime number p embeds in F.
- Def 9.5: Let F be a field. The prime field of F is the subfield
of F generated by 1_{F}.
- Examples
- Def 9.7: The (relative) degree of a field extension K/F,
denoted [K:F], is the dimension dim_{F}(K) of K as a vector
space over F.
An extension K/F is finite if [K:F] is finite, and is infinite
otherwise.
- Thm 9.8: (Degree Formula): Let F ⊆ K ⊆ L be field extensions.
Then [L:F] = [K:F][L:K]. Hence the composite of two finite extensions is a finite
extension.
- Examples
- Def 9.10: Let F ⊆ K be a field extension and let A ⊆ K.
Let S_{F,A} := {E ⊆ K | E is a field, F ⊆ E, and A ⊆ E}.
Then
F(A) := ∩_{E ∈ SF,A} E is the
subfield of K generated by A over F. In the case that
A = {a_{1},...,a_{m}} is finite, we denote
F(a_{1},...,a_{m}) := F(A).
- Def 9.11: A field extension K/F is simple if K = F(a)
for some a ∈ K. In this case a is a primitive element
for the extension K/F.
- Lemma 9.12: Let K/F be a field extension and let a ∈ K.
Then F(a) = {p(a)(q(a))^{-1} | p,q ∈ F[x], q ≠ 0}.
- Examples
- Algebraic and transcendental extensions
- Cor 9.15: Let F be a field and let p ∈ F[x] be an irreducible
polynomial of degree n, let K := F[x]/(p), and let α := x + (p).
Then K is an extension field of F of degree
n = degree(p) in which p has a root. Moreover,
K = {f_{0} + f_{1}α + ··· +
f_{n-1}α^{n-1} | f_{0},...,f_{n-1}
∈ F} is the set of all polynomials in α over F of degree < n,
with addition given by polynomial addition, and multiplication
defined by q(α)q'(α) = r(α) where r is the remainder
obtained after dividing the polynomial q(x)q'(x) by p(x) in F[x].
- Def 9.18: For a field extension K/F, an element a ∈ K is
algebraic over F if q(a) = 0 for some polynomial q ∈ F[x] of
positive degree; otherwise a is transcendental over F.
- Def 9.19: Let F be a field. The field of fractions of F[x] is denoted
F(x).
- Thm 9.20: Let K/F be a field extension and let a ∈ K.
Let I := {p ∈ F[x] | p(a) = 0}. Then
(1) I is an ideal of F[x].
(2) I = 0 if and only if a is transcendental.
(3) If a is algebraic over F then the unique monic generator of the
ideal I,
denoted m_{a,F} and called the minimum polynomial
of a over F, is irreducible.
(4) If a is algebraic over F then [F(a):F] = degree(m_{a,F}) and
there is an isomorphism
h: F[x]/(m_{a,F}) → F(a)
(isomorphic as fields) such that h(f +(m_{a,F})) = f for all f ∈ F
and h(x + (m_{a,F})) = a.
(5) If a is transcendental over F then there is an isomorphism
h: F(x) → F(a) such that h(f) = f for all f ∈ F and
h(x) = a.
(6) The element a is algebraic over F if and only if
[F(a):F] < ∞.
- Cor 9.21: If K/F is a finite field extension, then K is generated over F
by a finite set of algebraic elements over F.
- Cor 9.22: If K/F is a field extension and a ∈ K is algebraic over F,
then F(a) = F[a].
- Def 9.24: An extension K/F of fields is algebraic if every
element of K is algebraic over F.
- Prop 9.25: Let L/K and K/F be field extensions. Then
L/F is an algebraic field extension if and only if both L/K and K/F are algebraic
field extensions.
- Prop 9.27: Let L/F be a field extension. Then the set
K = {a ∈ L | a is algebraic over F} is a subfield of L that
contains F. Moreover, K is the largest subfield of L that is algebraic
over F.
- Examples
- Def 9.28: Q' := {a ∈ C | a is algebraic over Q}
is the field of algebraic numbers.
- Q' is an algebraic extension of Q,
and [Q' : Q] = ∞.
- Roots of unity
- Splitting fields
- Def 9.30: For a field F and a nonconstant polynomial p ∈ F[x],
a splitting field of p over F is an extension field K ⊇ F
such that p splits completely into linear factors in K[x], and p does
not split completely into linear factors over any proper subfield of K
that contains F.
- Examples
- Thm 9.33: (Existence of Splitting Fields): If F is a field and p ∈ F[x] is a nonconstant polynomial,
then there exists a splitting field for p over F.
- Prop 9.35: Let j: F → F' be an isomorphism of fields and let
p ∈ F[x] be a polynomial of positive degree. Let k: F[x] → F'[x] be
the induced ring homomorphism such that k(f) = j(f) for all f ∈ F
and k(x) = x, and let
p' := k(p) ∈ F'[x]. Let b be any root of p in an extension field L of F,
and let b' be any root of p' in an extension field L' of F'.
(1) If p is an irreducible polynomial in F[x], then there is a
field isomorphism h: F(b) → F'(b') such that h(f) = j(f) for all f ∈ F
and h(b) = b'.
(2) If K is a splitting field of p over F and K' is a splitting field
of p' over F', then there is a field isomorphism
h: K → K' such that h(f) = j(f) for all f ∈ F.
- Cor 9.37: Let F be a field, let p ∈ F[x].
(1) If p is an irreducible polynomial in F[x] and b,c are any two
roots of p in an extension field L of F, then there is a field isomorphism
h: F(b) → F(c) such that h|_{F} = Id_{F} and h(b) = c.
(2) (Uniqueness of Splitting Fields): If K,K' are any two splitting fields of p over F, then
there is a field isomorphism
h: K → K' such that h|_{F} = Id_{F}.
- Prop 9.40: Let F be a field, let p ∈ F be a polynomial
of positive degree n, and let K be a splitting field of p over F.
Then [K : F] ≤ n!.
- Examples
- Algebraic closure
- Def 9.43: A field L is algebraically closed if every nonconstant
polynomial p ∈ L[x] has a root in L[x].
- Rmk 9.44: L is algebraically closed if and only if every nonconstant
polynomial p ∈ L factors completely into linear factors.
- Examples
- Def 9.46: Let F be a field. An algebraic closure of F
is an algebraic extension field F' of F such that F' is algebraically closed.
- Thm 9.48: (Existence and Uniqueness of Algebraic Closure):
Let F be a field. Then there exists an algebraic closure of F.
Moreover, if L and L' are any two algebraic closures of F, then there is a
field isomorphism h: L → L' such that h|_{F} = Id_{F}.
- Prop 9.50: If L/F is an algebraic field extension and every nonconstant
polynomial p ∈ F[x] factors completely into linear factors in L[x],
then L is algebraically closed (and hence L is an algebraic closure of F).
- Example:
- Def 9.51: Let n ≥ 2 and let ζ_{n} :=
e^{2πi/n}. The n-th cyclotomic extension
is the extension Q(ζ_{n})/Q.
The primitive n-th roots of 1 are the powers
ζ_{n}^{k} for all 1 ≤ k ≤ n-1 such that
gcd(k,n)=1. The n-th cyclotomic polynomial Φ_{n} is
∏_{[1 ≤ k ≤ n-1 and gcd(k,n)=1]} (x - ζ_{n}^{k}).
The Euler totient function is the function φ: N → N
defined by φ(n) := |{k | 1 ≤ k ≤ n-1 and gcd(k,n)=1}|.
- Prop 9.52: (1) For all n ≥ 2,
Φ_{n} = m_{ζn,Q} ∈ Q[x] and
(hence) [Q(ζ_{n}) : Q] = φ(n).
(2) If Q' is the field of algebraic numbers, then Q' is
the algebraic closure of Q, Q' is algebraically closed,
and [Q' : Q] = ∞.
- Separability
- Def 9.55: Let F be a field, let q ∈ F[x], and let
b be a root of the polynomial q in a splitting field K of q over F.
The multiplicity of the root b
is the number of times (x-b) appears in a factorization of
q into linear factors in K[x]. If every root of q has multiplicity 1,
then the polynomial q is separable.
- Def 9.56: A field extension K/F is separable
if K/F is algebraic
and for all k ∈ K, the minimum polynomial m_{k,F} is separable.
- Example: Q(i) / Q
- Def 9.58: Let F be a field and let q = a_{n}x^{n} +
··· + a_{1}x + a_{0} ∈ F[x], with
each a_{i} ∈ F. The derivative of q is the polynomial
q' := na_{n}x^{n-1} +
··· + a_{1}.
- Lemma 9.59: Let F be a field and let p,q ∈ F[x] and r ∈ F.
(a) (rp)'=r(p'). (b) (p + q)' = p' + q'. (c) (pq)' = (p')q + p(q').
- Prop 9.60: Let F be a field and let q ∈ F[x]. (1) A root b of q has
multiplicity ≥ 2 if and only if b is a root of q'.
(2) The polynomial q is separable if and only if 1 is a gcd of q and q'.
(3) If q is irreducible in F[x], then q is separable if and only if q' ≠ 0.
- Examples: q = x^{3} - 1 in Z/3Z[x] and in R[x]
- Cor 9.62: Let F be a field with characteristic char(F) = 0.
(1) Every
irreducible polynomial in F[x] is separable.
(2) Every algebraic extension K/F is separable.
- Prop 9.63: Let F be a field with
characteristic char(F) = p for some prime number p, and let K/F be an
algebraic extension.
(1) If b is an element of F that is not a p-th power of an element of F,
and K/F is an algebraic extension of F that contains a root of x^{p} - b,
then K/F is not separable.
(2) If every element of
F is the p-th power of another element of F, then every algebraic
extension K/F is separable.
- Lemma 9.65: (Freshman's Dream):
Let R be a commutative ring with 1 ≠ 0 of characteristic char(R) = p
for some prime number p. Then the function h: R → R defined by
h(a) := a^{p} for all a ∈ R is a ring homomorphism.
- Cor 9.67: Every algebraic field extension of a finite field is separable.
- Section B: Galois Theory
- Group actions on field extensions
- Def 9.70: (a) Let K be a field. The automorphism group
of K, denoted Aut(K), is the collection of field automorphisms
of K, with the binary operation of
composition.
(b) Let K/F be a field extension. The automorphism group
of K/F, denoted Aut(K/F), is the collection of field automorphisms
of K that restrict to the identity on F, with the binary operation of
composition.
- Lemma 9.71: Let K/F be a field extension. Then Aut(K) is a group
and Aut(K/F) is a subgroup of Aut(K).
- Examples
- Lemma 9.73: Let K be a field, and let F be the prime field
of K. Then Aut(K/F) = Aut(K).
- Def 9.75: Let K be a field and let σ ∈ Aut(K). Then
for each q = a_{n}x^{n} + ··· + a_{0}
∈ K[x] (with each a_{i} ∈ K), let q^{σ}
denote the polynomial q^{σ} :=
σ(_{n})x^{n} + ··· + σ(a_{0}).
- Lemma 9.77: Let K/F be a field extension,
let σ ∈ Aut(K/F), and let q ∈ F[x].
(1) For all k ∈ K, q(σ(k)) = σ(q(k)).
(2) If b ∈ K is a root of q, then σ(b) also is a root of q.
- Thm 9.78: Let K/F be the splitting field of a polynomial q ∈ F[x].
Let Y be the set of distinct roots of q in K, and let n := |Y|.
(1) Aut(K/F) acts faithfully on Y, via σ · b := σ(b)
for all σ ∈ Aut(K/F) and b ∈ Y, and hence
Aut(K/F) is isomorphic to a subgroup of S_{n}.
(2) The orbits of the action of Aut(K/F) on Y are the subsets of Y
that are the roots of the same irreducible factor of q.
(3) If q is an irreducible polynomial in F[x], then
Aut(K/F) acts transitively on Y.
- Examples
- The splitting field of x^{3} - 2 over Q
- Interactions between group orders and field extension degrees
- Thm 9.80: Let K/F be a finite field extension. Then:
(1) |Aut(K/F)| ≤ [K : F].
(2) If K is the splitting field of a separable polynomial in F[x],
then |Aut(K/F)| = [K : F].
- Def 9.82:
Let K/F be a finite field extension. If |Aut(K/F)| = [K : F]
then the extension
K/F is called a Galois extension, and
the Galois group of K/F is Gal(K/F) := Aut(K/F).
- Examples
- Constructing Galois extensions, and the statement of Artin's Theorem
- Construction 1: Splitting fields of separable polynomials
- Def 9.84: Let F be a field and let q ∈ F[x] be a separable polynomial.
The Galois group of q over F is Gal(L/F), where
L is the splitting field of q over F.
- Construction 2: Fixed sets of subgroups of Aut(K)
- Def 9.86: Let K be a field and let G < Aut(K). The
subfield of K fixed by G
is K^{G} := {k ∈ K | g(k) = k for all g ∈ G}.
- Lemma 9.87: Let K be a field and let G < Aut(K). Then
K^{G} is a subfield of K.
- Thm 9.90: (Artin's Theorem): Let K be a field and let
G be a finite subgroup of Aut(K). Then K/K^{G} is a finite
Galois extension, and Gal(K/K^{G}) = G.
- Equivalence of the two constructions
- Cor 9.92: Let K/F be a finite Galois field extension. Then F = K^{Gal(K/F)}.
- Cor 9.94: Let K/F be a finite Galois field extension. For all k ∈ K,
the minimal polynomial m_{k,F} is separable and all of its roots are in K.
- Cor 9.95: Let K/F be a finite field extension. Then K/F is Galois
if and only if K is the splitting field of some separable polynomial in F[x].
- Examples
- Interactions between subgroups and "sub-(field extensions)"
- Def 9.97: Let L/F be a field extension. An intermediate field of L/F
is a subfield K of L that contains F; that is, F ⊆ K ⊆ L.
- Cor 9.98: Let L/F be a finite Galois field extension and let K be
an intermediate field of L/F. Then L/K is a finite Galois field extension.
- Thm 9.100: (Fundamental Theorem of Galois Theory (FTGT)): Let L/F
be a finite Galois field extension. Then the function
Ψ: {fields K | F ⊆ K ⊆ L} → {subgroups H of Gal(L/F)}
defined by Ψ(K) := Gal(L/K) is a bijection, with inverse defined by
Ψ^{-1}(H) := L^{H}. Moreover:
(1) Ψ and Ψ^{-1} each
reverse the order of inclusion.
(2) Ψ and Ψ^{-1} preserve index/degree; that is:
(2a) For all intermediate fields K of L/F,
[K : F] = |Gal(L/F) : Gal(L/K)|. (2b) For all subgroups H of Gal(L/F),
|Gal(L/F) : H| = [L^{H} : F].
(3) Galois extensions correspond to normal subgroups; that is:
(3a) If K is an intermediate field of L/F and L/K is Galois, then
Gal(L/K) ⊴ Gal(L/F).
(3b) If N ⊴ Gal(L/F), then L^{N}/F is Galois.
(4) If N ⊴ Gal(L/F), then
Gal(L^{N}/F) ≅ Gal(L/F)/N.
- Examples and applications
- Cyclotomic extensions revisited
- Def 9.105: Let F be a field and let n be
a positive integer such that char(F) does not divide n.
A primitive n-th root of 1 over F is an element ζ in
the splitting field K of x^{n} - 1 over F such that
ζ generates the (multiplicative) subgroup
H := {roots of x^{n} - 1 in K} ≤ K^{x}.
- Prop 9.107: Let F be a field, let n be
a positive integer such that char(F) does not divide n,
and let F' be the algebraic closure of F.
If ζ ∈ F' is a primitive n-th root of 1 over F, then F(ζ)/F
is a finite Galois extension, and Gal(F(ζ)/F) is an
abelian group that is isomorphic to a subgroup of
Z/nZ^{x}.
- Rmk: If F is a field of prime characteristic p
and n is an integer that is divisible by p,
then x^{n} - 1 is not separable.
- Splitting fields of polynomials of the form x^{n} - b
- Example: Computing the intermediate fields Q ⊆ K
⊆ L for the splitting field L of x^{4} - 2 over Q.
- Prop 9.110: Let F be a field and let n be
a positive integer such that char(F) does not divide n.
Let b ∈ F and let L be the splitting field of
x^{n} - b over F.
If F contains a primitive n-th root of 1,
then L/F is a finite Galois extension and
Gal(L/F) is isomorphic to a subgroup of Z/nZ
and hence is cyclic.
- Finite fields
- Thm 9.115: Let p be a prime number and let n be a
positive integer. Then the splitting field L of x^{pn} - x
over Z/pZ is a field of order p^{n}.
Moreover, if L' is another field of order p^{n}, then
L' ≅ L.
- Notation 9.117: Let p be a prime number and let n be a
positive integer. The unique (up to isomorphism) field of
order p^{n} is denoted F_{pn}.
- Thm 9.119: Let p be a prime number and let n be a
positive integer. Then the extension
F_{pn}/F_{p}
is Galois, and
Gal(F_{pn}/F_{p})
is a cyclic group of order n generated by the
element h ∈ Gal(F_{pn}/F_{p})
defined by h(k) := k^{p} for all
k ∈ F_{pn}.
- Proofs of Artin's Theorem and the Fundamental Theorem of Galois Theory
- Preliminaries
- Def 9.125: Let G be a group and let F be a field. A character
of G with values in F is a group homomorphism : G → F^{x}.
- Def 9.126: Let G be a group, let F be a field, and let
χ_{1},...,χ_{n} be characters of G with values in F.
These characters are linearly independent if
{χ_{1},...,χ_{n}} is a linearly independent
subset in the F-vector space of all functions : G → F^{x}.
- Thm 9.128: (Linear Independence of Characters):
Let G be a group, let F be a field, and
let χ_{1},...,χ_{m} : G → F^{x}
be distinct characters of G with values in F. Then
χ_{1},...,χ_{m} are linearly independent.
- Proof of Artin's Theorem
- Proof of FTGT
- Further examples and applications
S. Hermiller