Abstract Algebra: Theory and Applications · Problem 21.5.2

Finite Fields — Thomas Judson

Mathematics — Abstract Algebra · basis-field-extensions-degree

Problem

Find a basis for each of the following field extensions. What is the degree of each extension? (a) Q(3,6)\mathbb{Q}(\sqrt{3}, \sqrt{6}) over Q\mathbb{Q} (b) Q(23,33)\mathbb{Q}(\sqrt[3]{2}, \sqrt[3]{3}) over Q\mathbb{Q} (c) Q(2,i)\mathbb{Q}(\sqrt{2}, i) over Q\mathbb{Q} (d) Q(3,5,7)\mathbb{Q}(\sqrt{3}, \sqrt{5}, \sqrt{7}) over Q\mathbb{Q} (e) Q(2,23)\mathbb{Q}(\sqrt{2}, \sqrt[3]{2}) over Q\mathbb{Q} (f) Q(8)\mathbb{Q}(\sqrt{8}) over Q(2)\mathbb{Q}(\sqrt{2}) (g) Q(i,2+i,3+i)\mathbb{Q}(i, \sqrt{2}+i, \sqrt{3}+i) over Q\mathbb{Q} (h) Q(2+5)\mathbb{Q}(\sqrt{2}+\sqrt{5}) over Q(5)\mathbb{Q}(\sqrt{5}) (i) Q(2,6+10)\mathbb{Q}(\sqrt{2}, \sqrt{6}+\sqrt{10}) over Q(3+5)\mathbb{Q}(\sqrt{3}+\sqrt{5}) Hint. (a) {1,2,3,6}\{1, \sqrt{2}, \sqrt{3}, \sqrt{6}\}; (c) {1,i,2,2 i}\{1, i, \sqrt{2}, \sqrt{2}\,i\}; (e) {1,21/6,21/3,21/2,22/3,25/6}\{1, 2^{1/6}, 2^{1/3}, 2^{1/2}, 2^{2/3}, 2^{5/6}\}.

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