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1. (a) Suppose that α ∈ C satisfies the equation α 2 − 4α + 13 = 0. Show that Q(α) = Q(i). (b) Let F = Q(i, √ 3, ω) ⊂ C, where ω = e 2πi/3 . Find a basis for F as a Q-vector space. 2. Let F be a field with characteristic not equal to 2, and let E ⊃ F be an extension of F. Let a, b ∈ E be such that a 2 ∈ F and b 2 ∈ F. Show that: (a) ab ∈ F(a + b) (b) F(a, b) = F(a + b) 3. Let E = F2[X]/hX3 + X + 1i. For each a ∈ E find irr(a, F2), its irreducible polynomial over F2. 4. For each of the following polynomials in Q[X], determine [L : Q], where L ⊂ C is the splitting field of the polynomial. (The field L is the subfield Q(a1, . . . an) ⊂ C, where a1, . . . , an ∈ C are all the roots of f.) (a) X4 + X2 + 1 (b) X5 − 1 (c) X3 + X2 + 1

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Abstract Algebra Questions
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