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Dr. Marques Sophie Office 519 Algebra II Fall Semester 2013 marques@cims.nyu.edu 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