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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 1999 • Question 6

Question

Math 6310 • Algebra Fall 1999 • Question 6

Let \(E\) be a normal extension of \(\mathbb {Q}\) contained in \(\overline {\mathbb {Q}}\) (the algebraic closure of \(\mathbb {Q}\)), and let \(K\) be an extension of \(\mathbb {Q}\) contained in \(E\). Let \(K'\) be a subfield of \(\overline {\mathbb {Q}}\) which is isomorphic to \(K\). Show that \(K'\) is contained in \(E\) and the Galois groups of \(E/K\) and \(E/K'\) are conjugate in the Galois group of \(E/\mathbb {Q}\).

Raw LaTeX

Let \(E\) be a normal extension of \(\mathbb{Q}\) contained in \(\overline{\mathbb{Q}}\) (the algebraic
closure of \(\mathbb{Q}\)), and let \(K\) be an extension of \(\mathbb{Q}\) contained in \(E\). Let \(K'\)
be a subfield of \(\overline{\mathbb{Q}}\) which is isomorphic to \(K\). Show that \(K'\) is contained
in \(E\) and the Galois groups of \(E/K\) and \(E/K'\) are conjugate in the Galois group of
\(E/\mathbb{Q}\).

Topic Tags

Affine Algebraic Geometry and Residue Fields Ext Functors and Module Extensions Field Extensions Galois Theory