ASSIGNMENT 6
Due Date: Nov 24, 2010
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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 ].
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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).
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