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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Summer 2016 • Question 8

Question

Math 6310 • Algebra Summer 2016 • Question 8

Prove that the polynomial \(x^5-x-1\) has no root in \(\mathbb {F}_9\), and that it is irreducible over \(\mathbb {F}_3\). Determine the integers \(n\) for which \(x^5-x-1\) is irreducible over \(\mathbb {F}_{3^n}\).

Raw LaTeX

Prove that the polynomial \(x^5-x-1\) has no root in \(\mathbb{F}_9\), and that it is irreducible
over \(\mathbb{F}_3\). Determine the integers \(n\) for which \(x^5-x-1\) is irreducible over
\(\mathbb{F}_{3^n}\).

Topic Tags

Finite Fields Galois Groups of Polynomials Irreducible Polynomials Polynomial Rings and Ring Homomorphism Kernels