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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2000 • Question 10

Question

Math 6310 • Algebra Fall 2000 • Question 10

Suppose \(F\) is a field and \(K=F(\alpha )\) is a finite Galois extension of \(F\). Assume that the Galois group of \(K\) over \(F\) is cyclic generated by an automorphism \(\sigma \) where \(\alpha ^\sigma =\alpha +1\). Prove that \(F\) has characteristic \(p\ne 0\) and that \(\alpha ^p-\alpha \in F\).

Raw LaTeX

Suppose \(F\) is a field and \(K=F(\alpha)\) is a finite Galois extension of \(F\). Assume that the
Galois group of \(K\) over \(F\) is cyclic generated by an automorphism \(\sigma\) where
\(\alpha^\sigma=\alpha+1\). Prove that \(F\) has characteristic \(p\ne 0\) and that
\(\alpha^p-\alpha\in F\).

Topic Tags

Field Extensions Finite Fields Finite Group Theory Galois Theory