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
. 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
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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