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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2001 • Question 3

Question

Math 6310 • Algebra Fall 2001 • Question 3

Show that a polynomial \(f(X)\) of degree \(n\) over the field \(\mathbb {Z}/p\mathbb {Z}\) is irreducible if and only if \(f(X)\) is relatively prime to the polynomial \(X^{p^m}-X\) for all \(m\) less than or equal to \(n/2\).

Raw LaTeX

Show that a polynomial \(f(X)\) of degree \(n\) over the field \(\mathbb{Z}/p\mathbb{Z}\) is
irreducible if and only if \(f(X)\) is relatively prime to the polynomial \(X^{p^m}-X\) for all
\(m\) less than or equal to \(n/2\).

Topic Tags

Galois Groups of Polynomials Irreducible Polynomials Polynomial Rings and Prime Ideals Polynomial Rings and Ring Homomorphism Kernels