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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2002 • Question 2

Question

Math 6310 • Algebra Fall 2002 • Question 2

Let \(E\subset F\) be an extension of finite fields. Let \(q\) be the number of elements of \(E\), and let \(\phi \colon F\to F\) be defined by \(\phi (a)=a^q\). Prove that \(F\) is a Galois extension of \(E\) and that \(\phi \) generates its Galois group.

Raw LaTeX

Let \(E\subset F\) be an extension of finite fields. Let \(q\) be the number of elements of \(E\),
and let \(\phi\colon F\to F\) be defined by \(\phi(a)=a^q\). Prove that \(F\) is a Galois extension
of \(E\) and that \(\phi\) generates its Galois group.

Topic Tags

Field Extensions Finite Fields Finite Group Theory Galois Theory