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 rgg) = ∑g ∈ G rg for all ∑g ∈ G rgg ∈ RG.
• Ideals in (Z/2Z)C_3 ≅ Z[x]/(2,x3).
• Thm 5.136: (Binomial Theorem): Let R be a commutative ring with 1 ≠ 0. Then (r+s)n = ∑i=0n b(n,i)risn-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 | rn ∈ 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-1R [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-1R 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-1R.
• 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-1R is a commutative ring with 1 ≠ 0, and the function i: R → S-1R 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})-1R.
• 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-1R where S = {rn | 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-1R → 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-1R.
• 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 I1,...,In be ideals of R.
(1) The function h: R → (R/I1) × ··· × (R/In) defined by h(r) = (r+I1,...,r+In) for all r ∈ R is a ring homomorphism with kernel I1 ∩ ··· ∩ In.
(2) If Ii + Ij = R for all i ≠ j, then I1 ∩ ··· ∩ In = I1 ··· In, h is onto, and R/(I1 ··· In) ≅ (R/I1) × ··· × (R/In).
• 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 a0 > a1 > a2 > ···.
• Examples
• A = N0
• A = any set, f: A → N0, a < b iff f(a) < f(b).
• B = any finite set, A = B*, < is "shortlex order"
• B = {x1,...,xk}, A = {x1n1···xknk | n1,...,nkN0}, < 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 → N0 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-1Z
• 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 = p1 ··· pm with m ≥ 1 and each pi irreducible, and moreover whenever r = p1 ··· pm = q1 ··· qn with n ≥ 1 and each qj irreducible, then m = n and there is a permutation σ ∈ Sm such that for all i, pi and qσ(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 I1 ⊊ I2 ⊊ I3 ⊊ ...; 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[x1,...,xn] 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 = anxn + ··· + a0 with each ai ∈ R and an ≠ 0, then an is the leading coefficient of p and a0 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 Q2, 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 Q2.
• 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 z1 → z2 → z3 → ···
(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
• Z × Z, D6
• Def 6.99: Let F be a field, let R = F[x1,...,xn] be a polynomial ring with variables in the set A = {x1,...,xn}, 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 = {x1,...,xn} be a finite set, and let MM(A) = {x1a1 ··· xnan | a1,...,anN0} be the set of monic monomials over A. A Groebner basis in F[x1,...,xn] is a subset S ⊆ MM(A) × F[x1,...,xn] such that the rewritings of the form:
mu + q → mv+q whenever (u,v) ∈ S, m ∈ MM(A), and q ∈ R[x1,...,xn]
satisfy:
(1) "Termination": There is no infinite chain of rewritings z1 → z2 → z3 → ···
(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[x1,...,xn] is a Groebner basis S in F[x1,...,xn] 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[x1,...,xn] 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[x1,...,xn] 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 = (x2+5)
• R = (Z/2Z)[x], I = (x2+1,x3)
• R = (Z/2Z)[x,y], I = (x2y+xy2,x2+x+1)
• Thm 6.104: (Hilbert basis theorem): If R is a commutative Noetherian ring with 1 ≠ 0, then so is R[x1,...,xn] for any n > 0.
• Cor 6.106: Let F be a field and R = F[x1,...,xn]. 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 1R ≠ 0 and let M be an R-module. Then 0Rm = 0M and (-1R)m = -m for all m ∈ M.
• Def 7.5: Let R be a commutative ring with 1R ≠ 0. An R-algebra is a ring A with 1A ≠ 0 together with a ring homomorphism f: R → A such that f(1R) = 1A 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=1n jknk | 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 HomR(M,N) denotes the set of all R-module homomorphisms from M to N, and EndR(M) = End(M) denotes the set HomR(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) HomR(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 ∈ HomR(M,N), m ∈ M, and r ∈ R. (2) EndR(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 ∈ HomR(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 M1,...,Mn be R-modules. Their direct product module M1 × ··· Mn is the abelian group M1 × ··· Mn with ring action given by r(m1,...,mn) = (rm1,...,rmn)) for all r ∈ R and mi ∈ Mi.
(b) Let Mα be an R-module for all α in an index set J. The direct product of the R-modules Mα is the Cartesian product Π[α ∈ J] Mα with addition defined by (mα)α ∈ J+(nα)α ∈ J = (mα+nα)α ∈ J and R-action defined by r(mα)α ∈ J = (rmα)α ∈ J. The direct sum of the R-modules Mα is the R-submodule [α ∈ J] Mα of the direct product Π[α ∈ J] Mα given by ⊕[α ∈ J] Mα = {(mα)α ∈ J | mα = 0α 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 = {r1a1 + ··· + rnan | n ≥ 0 and each ri ∈ R and ai ∈ 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 {kxiyj | k ∈ Z, i,j ∈ N0} 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 FR(A) = F(A), the set of formal sums F(A) = {r1a1 + ··· + rnan | n ≥ 0 and each ri ∈ R and ai ∈ A} = { ∑[a ∈ A] raa | each ra ∈ R and ra = 0 for all but finitely many a}, with addition defined by (∑[a ∈ A] raa) + (∑[a ∈ A] saa) = ∑[a ∈ A] (ra + sa)a, and R-action defined by r(∑[a ∈ A] raa) = ∑[a ∈ A] (rra)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 FR(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 Rn = R × ··· ×R (with n factors).
• Def 7.41: Let M be an R-module. A linear combination of finitely many elements m1,...,mn of M is an element of M of the form r1m1 + ··· + rnmn for some r1,...,rn ∈ 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 r1,...,rn ∈ R and a1,...,an are distinct elements of A satisfying r1a1 + ··· + rnan = 0, then r1 = ··· = rn = 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 ≅ FR(B). (2) If and r1,...,rm,r1',...,rn' ∈ R - {0} and a1,...,am,a1',...,an' ∈ A satisfy ai ≠ aj and ai' ≠ aj' for all i ≠ j and r1a1 + ··· + rmam = r1'a1' + ··· + rn'an', then m = n and there is a permutation σ ∈ Sn such that ri = rσ(i) and ai = aσ(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 {e1,...,en} of Rn, where ei = (0,...,0,1,0,...,0) (with 1 in the i-th coordinate) is the standard basis of Rn.
• The empty set is linearly independent.
• Def 7.52: Let N be an R-module. The annihilator of N in R is annR(N) = {r ∈ R | rn = 0 for all n ∈ N}.
• Lemma 7.53: Let M be an R-module. (1) annR(M) is an ideal of R. (2) For any m ∈ M, the singleton set {m} is linearly independent if and only if annR(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 R2 and R3
• 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 ng > 0 such that gng = 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 = {c1,...,cn} be a linearly independent subset of V. Then there exist distinct elements b1,...,bn ∈ B such that (B - {b1,...,bn}) ∪ {c1,...,cn} 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 Fm for some m ≥ 0. (2) For any m,n ∈ N0, Fm ≅ Fn if and only if m = n.
• Cor 8.6: Let R be a commutative ring with 1 ≠ 0 and let m,n ∈ N. Then Rm ≅ Rn 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 = {b1,...,bn}, let W be an F-vector space of dimension m with ordered basis C = {c1,...,cm}, and let f: V → W be a linear transformation. Then [f]BC is the m × n matrix with entry tij ∈ F in row i column j determined by the equation f(bj) = ∑i=1m tijci.
• Example
• Def 8.21: Let R be a commutative ring with 1 ≠ 0 and let f: Rn → Rm be an R-module homomorphism. For each 1 ≤ j ≤ n let ej = (0,...,0,1,0,...,0) ∈ Rn, with the nonzero entry in the j-th coordinate, and for each 1 ≤ i ≤ m let ei' = (0,...,0,1,0,...,0) ∈ Rm, with the nonzero entry in the i-th coordinate. Then [f]=[f]{ej}{ei'} is the m × n matrix with entry tij ∈ F in row i column j (uniquely) determined by the equation f(bj) = ∑i=1m tijci.
• Def 8.23: Let R be a commutative ring with 1 ≠ 0. Let Mm,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 = {b1,...,bn}, let W be an F-vector space of dimension m with ordered basis C = {c1,...,cm}, and let f: V → W be a linear transformation. Then the function h=hB,C: HomF(V,W) → Matm,n(F) defined by h(f) = [f]BC 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]BD = [g]CD[f]BC (where the latter product is the usual matrix multiplication).
• Def 8.27: Let F be a field. A matrix P ∈ Mn,n(F) is invertible if there is a matrix Q ∈ Mn,n(F) satisfying PQ = In = QP. The general linear group GLn(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=hB,B : HomF(V,V) → Matn,n(F), defined by h(f) = [f]BB for all f ∈ HomF(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) → GLn(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 = {b1,...,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 bi by bi + rbj for some i ≠ j and some r ∈ R. (2) Replacing bi by ubi for some i and some unit u of R. (3) Swapping the indices of bi and bj for some i ≠ j.
• Def 8.31: Let R be a commutative ring with 1 ≠ 0. An elementary row operation on a matrix A ∈ Matm,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 In by applying a single elementary row operation.
(1) For r ∈ R and 1 ≤ i,j ≤ n with i ≠ j, let Eij(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 ∈ Rx and 1 ≤ i ≤ n let Ei(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 ∈ Matn,q(R), then the product matrix EA is the result of performing the corresponding elementary row operation on A. If B ∈ Matm,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 IdV be the identity map : V → V. Then [IdV]BB' 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' = [IdW]CC' [f]BC [IdV]B'B. (3) Let g : V → V be a linear transformation. Then [g]BB and [g]B'B' are similar.
• Def 8.41: Let F be a field. The column rank of a matrix M ∈ Matm,n(F) is the maximum number of linearly independent columns of M (viewed as elements of Fm).
• Prop 8.42: Let F be a field, let f: Fn → Fm, and let B = {b1,...,bn} and C = {c1,...,cm} be the standard ordered bases of Fn and Fm, respectively. Then rank(f) equals the column rank of the matrix [f]BC.
• Def 8.43: Let F be a field and let A ∈ Matm,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 aiji of the i-th row of A' is 1, all other entries in the ji-th column are 0, and j1 < j2 < ··· jr, where r is the number of nonzero rows of A'. Then numbers ji are the pivots of A'.
• Prop 8.44: Let F be a field, let f: Fn → Fm, and let B = {b1,...,bn} and C = {c1,...,cm} be the standard ordered bases of Fn and Fm, respectively. If A' is an rref of [f]BC with pivots j1,...,jr, then {f(bj1),...,f(bjr)} is a basis for f(Fn).
• Examples
• Determinants
• Def 8.45: Let R be a commutative ring with 1 ≠ 0. For any v1,...,vn ∈ Rn, let [v1 | ··· | vn] denote the n × n matrix with the vi's as columns.
• Def 8.46: Let R be a commutative ring with 1 ≠ 0. A determinant function is a function d : Matn,n(R) → R for some n ≥ 1 satisfying the following: (1) d is multilinear on columns; that is, whenever 1 ≤ i ≤ n, v1,...,vn,vi' ∈ Rn, and r ∈ R, then d([v1 | ··· | vi-1 | vi+vi' | vi+1 | ··· | vn]) = d([v1 | ··· | vn]) + d([v1 | ··· | vi-1 | vi' | vi+1 | ··· | vn]) and d([v1 | ··· | vi-1 | rvi | vi+1 | ··· | vn]) = rd([v1 | ··· | vn]) (2) d is alternating; that is, whenever vi = vj for some i ≠ j, then d([v1 | ··· | vn]) = 0. (3) d(In) = 1.
• Prop 8.48: Let R be an integral domain and let d : Matn,n(R) → R be a determinant function, and let M ∈ Matn,n(R).
(1) For any r ∈ R and 1 ≤ i,j ≤ n with i ≠ j, d(M Eij(r)) = d(M).
(2) For any u ∈ Rx and 1 ≤ i ≤ n, d(M Ei(u)) = u d(M).
(3) For any 1 ≤ i,j ≤ n with i ≠ j, d(M E(i j)) = -d(M).
(4) If v1,...,vn ∈ Rn are linearly dependent, then d([v1 | ··· | vn]) = 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 : Matn,n(R) → R is given by det([aij]) = ∑[σ ∈ Sn] sign(σ) ∏i=1n aiσ(i).
• Rmk: The function det is a (homogeneous) polynomial in the aij's of degree n, with n! terms.
• Prop 8.52: Let R be a commutative ring with 1 ≠ 0. If A,B ∈ Matn,n(R) then det(AB) = det(A) det(B).
• Thm 8.53: (1) Let F be a field and let A ∈ Matn,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 ∈ Matn,n(R). Then A is invertible if and only if det(A) ∈ Rx.
• 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]BB) 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 ∈ Matn,n(F). A nonzero element v ∈ Fn 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 λ1,...,λk are distinct eigenvalues of t, and for each 1 ≤ i ≤ k, vi is an eigenvector of t with eigenvalue λi, then {v1,...,vk} 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 ∈ Matm,n(R), and let tA: Rn → Rm be the R-module homomorphism represented by A over the standard bases. The R-module presented by A is the R-module Rm/tA(Rn).
• 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 f1: X1 → X2, f2 : X2 → X3,... is exact if for each index i, fi(Xi) = Ker(fi+1).
• The classification theorem and Smith Normal Form
• Example
• Prop 8.67: Let R be a commutative ring with 1 ≠ 0 and let A ∈ Matm,n(R) and B ∈ Matp,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 Aej = ei (where ei denote the i-th standard basis vector in any Rk). (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 ∈ Matm,n(R). Then there is a sequence of elementary row and column operations that transform A into a matrix A' = [aij'] such that all non-diagonal entries of A' are 0, and the diagonal entries of A' satisfy a11' | a22' | a33' | ···. Moreover, the number j of nonzero entries of A' is uniquely determined by A, and the nonzero diagonal entries a11',...,ajj' are unique up to associates.
• Lemma 8.72: Let R be a commutative ring with 1 ≠ 0, let m ≥ n, let A = [aij] ∈ Matm,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/(a11) ⊕ ··· ⊕ R/(ann) ⊕ Rm-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 d1,...,dk of R satisfying di | di+1 for all i, such that M ≅ Rr ⊕ R/(d1) ⊕ ··· ⊕ R/(dk). Moreover r and k are uniquely determined by M, and the di are unique up to associates.
• Def 8.76: Let R be a PID, let r ≥ 0, k ≥ 0, and let d1,...,dk be nonzero nonunit elements of R satisfying di | di+1 for all i. Let M be the R-module M ≅ Rr ⊕ R/(d1) ⊕ ··· ⊕ R/(dk). The free rank of M is the integer r. The elements d1,...,dk 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 p1,...,ps of R, and e1,...,es ≥ 1, such that M ≅ Rr ⊕ R/(p1e1) ⊕ ··· ⊕ R/(pses). Moreover r and k are uniquely determined by M, and the list p1e1,..., pses is unique up to associates and reordering.
• Def 8.78: Let R be a PID, let r ≥ 0, s ≥ 0, let p1,...,ps be prime elements of R, and let e1,...,es ≥ 1. Let M be the R-module M ≅ Rr ⊕ R/(p1e1) ⊕ ··· ⊕ R/(pses). The elements p1e1,..., pses 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 Vt is defined to be the vector space V with the unique F[x]-action satisfying xv = t(v) for all v ∈ V. [That is, (rnxn + ··· + r0)v = rntn(v) + ··· + r0v for all rnxn + ··· + r0 ∈ 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 g1,...,gk ∈ F[x] of positive degree such that gi | gi+1 for all i and there is an F[x]-module isomorphism Vt ≅ F[x]/(g1) ⊕ ··· ⊕ F[x]/(gk). Moreover, k and the gi are uniquely determined by V.
• Def 8.85: The polynomials g1,...,gk in Prop 8.84 are the invariant factors of the linear transformation t.
• Def 8.87: Let F be a field and let p = xn + rn-1xn-1 + ··· + r0 ∈ F[x] be a monic polynomial. The companion matrix of p, denoted C(p), is the matrix [cij] in Mn,n(F) satisfying ci,i-1 = 1 for all 2 ≤ i ≤ n, cin = -ri-1 for all 1 ≤ i ≤ n, and cij = 0 for all other i,j.
• Def 8.88: Let F be a field and let A ∈ Matm,m(F) and B ∈ Matn,n(F). The matrix A ⊕ B is the matrix [cij] in Mm+n,m+n(F) defined by cij = aij for all 1 ≤ i,j ≤ m, cij = bi-m,j-m for all m+1 ≤ i,j ≤ m+n, and cij = 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]BB = C(g1) ⊕ ··· ⊕ C(gk) for some monic polynomials g1,...,gk ∈ F[x] of positive degree satisfying gi | gi+1 for all i. Moreover, the gi 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 g1,...,gk (with gi | gi+1 for all i). The matrix C(g1) ⊕ ··· ⊕ C(gk) is the rational canonical form of t. (2) Let A ∈ Matn,n(F), and let t: Fn → Fn be the linear transformation such that A = [t]EE, where E is the standard basis of Fn. The rational canonical form of A is the rational canonical form of tA.
• 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]BB. Then the matrix xIn - A ∈ Matn,n(F[x]) presents the F[x]-module Vt.
• 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]BB. The invariant factors of t are the nonzero nonunit entries in the Smith normal form of the matrix xIn - A.
• Def 8.99: Let F be a field. (1) Let A ∈ Matn,n(F). The characteristic polynomial of A, denoted cA, is the polynomial cA = det(xIn - 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 ct, is the polynomial cA for a matrix A = [t]BB 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 ct of t is well-defined. (2) For any A ∈ Matm,m(F) and B ∈ Matn,n(F), cA ⊕ B = cAcB.
• Def 8.102: Let F be a field. (1) Let A ∈ Matn,n(F). A minimal polynomial of A, denoted mA, is a monic polynomial of least degree such that mA(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 mt, is the monic polynomial generating the annihilator ideal annF[x](Vt) 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 mA of A = [t]BB satisfies mA = mt, and hence is independent of the choice of ordered basis B of V. (2) For any polynomial p ∈ F[x], mC(p) = cC(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 g1,...,gk (with gi | gi+1 for all i).
(1) ct = g1 ··· gk.
(2) mt = gk.
(3) (Cayley-Hamilton Thm): mt | ct, and hence ct(t) = 0.
(4) ct | mtk.
(5) Let f ∈ F. The following are equivalent: (5a) f is an eigenvalue of t. (5b) f is a root of mt. (5c) f is a root of ct.
• Thm 8.107: Let F be a field and let A,A' ∈ Matn,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) xIn - A and xIn - 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 Matn,n(K).
• Jordan Canonical Form
• Def 8.110: Let F be a field, let n > 0, and let r ∈ F. The Jordan block Jn(r) is the n × n matrix over F with entries aij (in row i column j) satisfying aii = r for all 1 ≤ i ≤ n, ai,i+1 = 1 for all 1 ≤ i ≤ n-1, and aij = 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 ct of t factors completely into linear terms. Then there is an ordered basis B for V such that [t]BB = Je1(r1) ⊕ ··· ⊕Jes(rs) such that each ri ∈ F is a root of ct and each ei ≥ 1. Moreover, the polynomials (x - r1)e1,...,(x - rs)es are the elementary divisors of the F[x]-module Vt, and this expression for [t]BB 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 - r1)e1,...,(x - rs)es. The matrix Je1(r1) ⊕ ··· ⊕Jes(rs) is a Jordan canonical form of t. (2) Let A ∈ Matn,n(F), and let t: Fn → Fn be the linear transformation such that A = [t]EE, where E is the standard basis of Fn. A Jordan canonical form of A is a Jordan canonical form of tA.
• 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]BB is a diagonal matrix. (2) Let A ∈ Matn,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 1F.
• Examples
• Def 9.7: The (relative) degree of a field extension K/F, denoted [K:F], is the dimension dimF(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 SF,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 = {a1,...,am} is finite, we denote F(a1,...,am) := 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 = {f0 + f1α + ··· + fn-1αn-1 | f0,...,fn-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 ma,F and called the minimum polynomial of a over F, is irreducible. (4) If a is algebraic over F then [F(a):F] = degree(ma,F) and there is an isomorphism h: F[x]/(ma,F) → F(a) (isomorphic as fields) such that h(f +(ma,F)) = f for all f ∈ F and h(x + (ma,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 = IdF 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 = IdF.
• 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 = IdF.
• 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 ζn := e2πi/n. The n-th cyclotomic extension is the extension Qn)/Q. The primitive n-th roots of 1 are the powers ζnk for all 1 ≤ k ≤ n-1 such that gcd(k,n)=1. The n-th cyclotomic polynomial Φn is ∏[1 ≤ k ≤ n-1 and gcd(k,n)=1] (x - ζnk). The Euler totient function is the function φ: NN defined by φ(n) := |{k | 1 ≤ k ≤ n-1 and gcd(k,n)=1}|.
• Prop 9.52: (1) For all n ≥ 2, Φn = mζn,QQ[x] and (hence) [Qn) : 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 mk,F is separable.
• Example: Q(i) / Q
• Def 9.58: Let F be a field and let q = anxn + ··· + a1x + a0 ∈ F[x], with each ai ∈ F. The derivative of q is the polynomial q' := nanxn-1 + ··· + a1.
• 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 = x3 - 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 xp - 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) := ap 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
• Aut(C/R)
• Aut(Q(√2)/Q)
• 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 = anxn + ··· + a0 ∈ K[x] (with each ai ∈ K), let qσ denote the polynomial qσ := σ(n)xn + ··· + σ(a0).
• 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 Sn. (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 x3 - 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
• Cyclotomic extensions
• 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 KG := {k ∈ K | g(k) = k for all g ∈ G}.
• Lemma 9.87: Let K be a field and let G < Aut(K). Then KG 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/KG is a finite Galois extension, and Gal(K/KG) = G.
• Equivalence of the two constructions
• Cor 9.92: Let K/F be a finite Galois field extension. Then F = KGal(K/F).
• Cor 9.94: Let K/F be a finite Galois field extension. For all k ∈ K, the minimal polynomial mk,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) := LH. 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| = [LH : 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 LN/F is Galois.
(4) If N ⊴ Gal(L/F), then Gal(LN/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 xn - 1 over F such that ζ generates the (multiplicative) subgroup H := {roots of xn - 1 in K} ≤ Kx.
• 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/nZx.
• Rmk: If F is a field of prime characteristic p and n is an integer that is divisible by p, then xn - 1 is not separable.
• Splitting fields of polynomials of the form xn - b
• Example: Computing the intermediate fields Q ⊆ K ⊆ L for the splitting field L of x4 - 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 xn - 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 xpn - x over Z/pZ is a field of order pn. Moreover, if L' is another field of order pn, 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 pn is denoted Fpn.
• Thm 9.119: Let p be a prime number and let n be a positive integer. Then the extension Fpn/Fp is Galois, and Gal(Fpn/Fp) is a cyclic group of order n generated by the element h ∈ Gal(Fpn/Fp) defined by h(k) := kp for all k ∈ Fpn.
• 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 → Fx.
• Def 9.126: Let G be a group, let F be a field, and let χ1,...,χn be characters of G with values in F. These characters are linearly independent if {χ1,...,χn} is a linearly independent subset in the F-vector space of all functions : G → Fx.
• Thm 9.128: (Linear Independence of Characters): Let G be a group, let F be a field, and let χ1,...,χm : G → Fx be distinct characters of G with values in F. Then χ1,...,χm are linearly independent.
• Proof of Artin's Theorem
• Proof of FTGT
• Further examples and applications

S. Hermiller