Math 547, Final Exam, Spring , 2005
The exam is worth 100 points. Each problem is worth 11 1/9 points.
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Take enough space for each problem. Turn in
your solutions in the order: problem 1, problem 2, ... ; although, by using enough
paper, you can do the problems in any order that suits you.
I will e-mail your grade to you as soon as I finish grading the exams.
I will post the solutions on my website later today.
1. Let K⊆Lbe fields, f(x) be a polynomial in K[x] , σ∈AutKL, and `∈L.
Suppose that f(`) = 0 . Prove f(σ(`)) = 0 . Give all details.
2. Let K⊆Lbe fields, f(x) be an irreducible polynomial of K[x] , and α1
and α2be elements of Lwith f(α1) = f(α2) = 0 . Prove that there exists a
ring isomorphism σ:K[α1]→K[α2] with σ(α1) = α2and σ(k) = kfor all
k∈K. Give all details.
3. State the Fundamental Theorem of Galois Theory. Please give hypotheses and
conclusions.
4. Let Ibe an ideal of the ring R. Prove that Iis a maximal ideal of Rif and
only if R/I is a field.
5. Prove that Q[x] is a Principal Ideal Domain.
6. Let Ibe an ideal in a Principal Ideal Domain R. Prove that the following
statements are equivalent. (That is, if one of the statements is true, then they
all are true. If one of the statements is false, then they all are false.)
(a) There is an irreducible element rof Rwith I= (r) .
(b) The ideal Iis a non-zero prime ideal.
(c) The ideal Iis a maximal ideal.
7. Let Kbe the splitting field of x5−2 over Q. We have shown that
K=Q[5
√2, ω] , where ω=e2πi
5. We have also shown that dimQK= 20 ,
and that there exist auotmorphisms σ, τ in AutQKwith
σ(5
√2) = 5
√2σ(ω) = ω2
τ(5
√2) = ω5
√2τ(ω) = ω.
Furthermore we have shown that AutQKis generated by σand τ. You do
not have to re-prove any of the above facts. However, I do want complete details
for the following things: Find a field Ewith Q⊆E⊆Kand dimQE= 2 .
Find the subgroup Hof AutQKwith KH=E. (“Find” means tell me
generators.)
