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ASSIGNMENT 6 Due Date: Nov 24, 2010 n 1) Prove that xp − x + 1 is irreducible over the finite field Fp only when n = 1 or n = p = 2. (Hint: Note that if α is a root, then so is α + a for any a in Fpn ). 2) Let f (x) ∈ F [x] be an irreducible polynomial of degree n over the field F , and let L be the splitting field of f (x) over F with α a root of f (x) in L. If K is any Galois extension of F , show that the polynomial f (x) splits into a product of m irreducible polynomials each of degree d over K, where d = [K(α) : K] = [(L∩K)(α) : L∩K] and m = n/d = [F (α)∩K : F ]. 4 2 3) Determine the √ group of f (x) = x − 10x + 1 ∈ Q[x]. (Hint: f (x) is the minimal √ Galois polynomial of 2 + 3 over Q). 4) Determine the splitting field of the polynomial xp − x − a over Fp , where a 6= 0, a ∈ Fp . Show explicitly that the Galois group is cyclic. (Such an extension is called an Artin-Schreier extension). 5) Prove using field theory that n | φ(pn − 1), where φ is Euler’s φ-function. (Hint: φ(pn − 1) is the order of the group of automorphisms of a cyclic group of order pn − 1. 6) Show that p(x) = x3 − 2x − 2 is irreducible over Q and let θ be a root. Compute (1 + θ)(1 + θ + θ2 ) in Q(θ). 1