# Math 817: Problem set 5

*Instructions*: Write up any four of the following problems (but
you should *try* all of them, and
in the end you should be sure you know how to do all of them).
Your write ups are due Friday, October 1, 2004.
Each problem is worth 10 points, 9 points for correctness
and 1 point for communication. (Your goal is not only
to give correct answers but to communicate
your ideas well. Make sure you use good English,
so proofread your solutions.
Once you finish a solution, you
should restructure awkward sentences, and strike out
anything that is not needed in your approach to the problem.)
- Let e denote the base of the natural logs (as usual).
You may assume the facts that e is transcendental (i.e.,
there is no nontrivial polynomial f(x) with rational
coefficients for which f(e) = 0) and that the
square root of 2 is not a rational number.
Let s denote the square root of 2. Regarding
the reals
**R** as a vector space over the rationals
**Q**, then V = Span({1, e}) and W = Span({1,s})
are **Q**-subspaces. With respect to the usual
operation of multiplication in **R**,
prove that W is in fact a field,
but that V is not.
- Prove that every solution of
2x
_{1} - x_{2} - 2x_{3} = 0
has the form (1.5) on p. 80. I.e., every solution is a linear combination
of **w**_{1} and **w**_{2}, where the components
of the former are 1, 0 and 1, and those of the latter are 1, 2 and 0.
- Define a concept of homomorphism of fields in such a way
that if f: F -> K is a homomorphism of fields, then
f is injective and Im(f) is a field. Your definition may not
explicitly include the requirement that f be injective or that
Im(f) be a field. Then prove that f is injective and Im(f)
is a field using your definition.
- Let A be the 2x2 matrix
8e
_{1,1} + 3e_{1,2} + 2e_{2,1} + 6e_{2,2},
let **x** be the 2x1 column vector x_{1}e_{1,1}
+ x_{2}e_{2,1}
and let **b** be the 2x1 column vector 3e_{1,1}
- e_{2,1}. (Compare #10, p. 104.)
- (a) Solve A
**x** = **b** in **F**_{p}, for
p = 2, 5, and 7.
- (b) Determine the number of solutions when p = 7.

- Let p be a positive integer which is a prime.
(See #14, p. 105.)
- (a) Let n be any integer. If p does not divide n,
prove that n
^{p-1} = 1 (mod p).
- (b) Let n be any integer. Prove that
n
^{p} = n (mod p).

- Let p be a positive integer which is a prime.
(See #15, p. 105.)
- (a) Prove that the product of all nonzero elements of
**F**_{p} is -1.
- (b) Prove Wilson's Theorem that (p - 1)! = -1 (mod p).