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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2009 • Question 3

Question

Math 6310 • Algebra Fall 2009 • Question 3

Let \(E\subset F\) be a finite Galois extension of fields. Suppose that there is an element \(\alpha \) in \(F\) such that \(\alpha \notin E\) and such that \(\alpha \) is in every proper extension of \(E\) contained in \(F\). Show that the Galois group of \(F\) over \(E\) is cyclic of prime power order.

Raw LaTeX

Let \(E\subset F\) be a finite Galois extension of fields. Suppose that there is an element
\(\alpha\) in \(F\) such that \(\alpha\notin E\) and such that \(\alpha\) is in every proper extension
of \(E\) contained in \(F\). Show that the Galois group of \(F\) over \(E\) is cyclic of prime power
order.

Topic Tags

Field Extensions Finite Fields Finite Group Theory Galois Theory