Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2016 • Question 8

Question

Math 6310 • Algebra Spring 2016 • Question 8

Let \(p\) be a prime number. Prove that if \(x^{p^n}-x+1\) is irreducible over \(\mathbb {F}_p\), then \(n=1\) or \(n=2=p\).

(Hint: Note that if \(\alpha \) is a root of the equation, then \(\alpha ^p-\alpha \) is in \(\mathbb {F}_{p^n}\).)

Raw LaTeX

Let \(p\) be a prime number. Prove that if \(x^{p^n}-x+1\) is irreducible over \(\mathbb{F}_p\),
then \(n=1\) or \(n=2=p\).

(Hint: Note that if \(\alpha\) is a root of the equation, then \(\alpha^p-\alpha\) is in
\(\mathbb{F}_{p^n}\).)

Topic Tags

Finite Fields Irreducible Polynomials