Abstract Algebra: Theory and Applications · Problem 23.5.1

Galois Theory — Thomas Judson

Mathematics — Abstract Algebra · galois-theory-galois-groups

Problem

Compute each of the following Galois groups. Which of these field extensions are normal field extensions? If the extension is not normal, find a normal extension of Q\mathbb{Q} in which the extension field is contained. (a) G(Q(30)/Q)G(\mathbb{Q}(\sqrt{30})/\mathbb{Q}) (b) G(Q(54)/Q)G(\mathbb{Q}(\sqrt[4]{5})/\mathbb{Q}) (c) G(Q(2,3,5)/Q)G(\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})/\mathbb{Q}) (d) G(Q(2,23,i)/Q)G(\mathbb{Q}(\sqrt{2}, \sqrt[3]{2}, i)/\mathbb{Q}) (e) G(Q(6,i)/Q)G(\mathbb{Q}(\sqrt{6}, i)/\mathbb{Q}) Hint. (a) Z2\mathbb{Z}_2; (c) Z2×Z2×Z2\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2.

Related theory

Browse the problem library