Abstract Algebra: Theory and Applications · Problem 23.5.2

Galois Theory — Thomas Judson

Mathematics — Abstract Algebra · galois-theory-separability

Problem

Determine the separability of each of the following polynomials. (a) x3+2x2−x−2x^3 + 2x^2 - x - 2 over Q\mathbb{Q} (b) x4+2x2+1x^4 + 2x^2 + 1 over Q\mathbb{Q} (c) x4+x2+1x^4 + x^2 + 1 over Z3\mathbb{Z}_3 (d) x3+x2+1x^3 + x^2 + 1 over Z2\mathbb{Z}_2 Hint. (a) Separable over Q\mathbb{Q} since x3+2x2−x−2=(x−1)(x+1)(x+2)x^3 + 2x^2 - x - 2 = (x-1)(x+1)(x+2); (c) not separable over Z3\mathbb{Z}_3 since x4+x2+1=(x+1)2(x+2)2x^4 + x^2 + 1 = (x+1)^2(x+2)^2.

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