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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2006 • Question 8

Question

Math 6310 • Algebra Spring 2006 • Question 8

Let \(K\) and \(F\) be fields such that \(K\) is a finite extension of \(F\). Assume that there are only finitely many subfields \(L\) such that \(F\subset L\subset K\). Show that \(K\) is a primitive extension of \(F\) (i.e. \(K=F(\theta )\) for some \(\theta \in K\)).

Raw LaTeX

Let \(K\) and \(F\) be fields such that \(K\) is a finite extension of \(F\). Assume that there are only
finitely many subfields \(L\) such that \(F\subset L\subset K\). Show that \(K\) is a primitive
extension of \(F\) (i.e. \(K=F(\theta)\) for some \(\theta\in K\)).

Topic Tags

Ext Functors and Module Extensions Field Extensions Finite Fields