Problem Set # 10

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Dr. Marques Sophie
Office 519
Algebra II
Fall Semester 2013
[email protected]
Problem Set # 10
Due Wednesday november 25th in recitation
Exercise 1(?): 100 points
Let F = (Z/3Z)[x]/(x2 + 1).
1. List the elements of F .
2. Find a generator g of F ∗ .
3. For each α ∈ F find the minimal polynomial pα (y) of α in (Z/3Z)[y].
4. Factor y 9 − y in F [y].
5. Factor y 9 − y in (Z/3Z)[y]. Show how the factors in F [y] join to form factors in
(Z/3Z)[y].
√
√
Exercise 2(?): 60 points Let K = Q( a1 , . . . , as ) with a1 , . . . , as ∈ Q.
1. Give an upper bound on [K : Q].
2. Let σ ∈ Gal(K, Q). Prove σ 2 = e.
3. Let σ, τ ∈ Gal(K, Q). Prove στ = τ σ.
Exercise 3(?): 40 points
Let p(x) √
∈ Q[x] be an irreducible cubic
C. Let K = Q(α, β, γ).
√ with roots α, β, γ ∈ √
Suppose a ∈ K where a ∈ Q and a 6∈ Q. Set L = Q( a). Prove p(x) is still
irreducible when considered in L[x]. (Hint: When cubics reduce they have a root.)
1
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