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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2018 • Question 10

Question

Math 6310 • Algebra Spring 2018 • Question 10

Let \(F\) be a field and \(n\) a positive integer such that \(F\) has no nontrivial field extensions of degree less than \(n\). Let \(L=F[\alpha ]\) be an extension such that \(\alpha ^n\) is in \(F\). Prove that each element of \(L\) is a product of elements of the form \(a\alpha +b\), with \(a,b\) in \(F\).

Raw LaTeX

Let \(F\) be a field and \(n\) a positive integer such that \(F\) has no nontrivial field extensions
of degree less than \(n\). Let \(L=F[\alpha]\) be an extension such that \(\alpha^n\) is in \(F\).
Prove that each element of \(L\) is a product of elements of the form \(a\alpha+b\), with \(a,b\) in
\(F\).

Topic Tags

Ext Functors and Module Extensions Field Extensions Tensor Products of Modules