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By David B Surowski

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Let p be a prime and let r be a positive integer. Prove that there exists an irreducible polynomial of degree r over Fp . 8. Let p be prime, n a positive integer and set q = pn . If f (x) ∈ Fp [x] is irreducible of degree m, show that f (x)|xq − x. More generally, show that if f (x) is irreducible of degree n, where n|m, then again, f (x)|xq − x. 4. SEPARABILITY AND THE GALOIS CRITERION 59 9. Let f (x) ∈ F[x] and assume that f (xn ) is divisible by (x − a)k , where 0 = a ∈ F. Prove that f (xn ) is also divisible by (xn − an )k .

K}, form the integer qM = √ i∈M pi . Thus the field KM = Q( qM ) is a subfield of E; prove that if M1 = M2 then KM1 = KM2 . Since there are 2k − 1 nonempty subsets of {1, 2, . . ) 10. Retain the notation and assumptions of the above exercise. Prove that √ √ √ √ √ √ Q( p1 + p2 + . . + pk ) = Q( p1 , p2 , . . , pk ). 11. ) Let F be a field and let x be indeterminate over F. Set E = F(x), a simple transcendental extension of F. (i) Let α ∈ E; thus α = f (x)/g(x), where f (x), g(x) ∈ F[x], and where f (x) and g(x) have no common factors.

Note that the algebraic closure of the field F, whose existence is guaranteed by the above theorem, is essentially unique (in the sense of Corollary 10, above). 2. 2 1. Let f (x) = xn − 1 ∈ Q[x]. In each case below, construct a splitting field K over Q for f (x), and compute [K : Q]. (i) n = p, a prime. (ii) n = 6. (iii) n = 12. Any conjectures? We’ll discuss this problem in Section 8. 2. Let f (x) = xn − 2 ∈ Q[x]. Construct a splitting field for f (x) over Q. ) 3. Let f (x) = x3 + x2 − 2x − 1 ∈ Q[x].

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