Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2010 • Question 9

Question

Math 6310 • Algebra Spring 2010 • Question 9

Let \(F\) be a field that contains a primitive \(n\)th root of unity. Show that if \(a\) is an element of \(F\) and the field \(E\) is obtained from \(F\) by adjoining an \(n\)th root of \(a\), then \(E\) is a Galois extension of \(F\) with cyclic Galois group.

Raw LaTeX

Let \(F\) be a field that contains a primitive \(n\)th root of unity. Show that if \(a\) is an element
of \(F\) and the field \(E\) is obtained from \(F\) by adjoining an \(n\)th root of \(a\), then \(E\) is a
Galois extension of \(F\) with cyclic Galois group.

Topic Tags

Cyclotomic Fields Ext Functors and Module Extensions Field Extensions Galois Theory