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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2016 • Question 7

Question

Math 6310 • Algebra Spring 2016 • Question 7

Let \(K\) be a field and \(f(x)\in K[x]\) an irreducible polynomial. Let \(n\ge 2\) be an integer and set \(g(x):=f(x^n)\). Prove that if \(h(x)\) is an irreducible factor of \(g(x)\), then the degree of \(f\) divides the degree of \(h\).

Raw LaTeX

Let \(K\) be a field and \(f(x)\in K[x]\) an irreducible polynomial. Let \(n\ge 2\) be an integer
and set \(g(x):=f(x^n)\). Prove that if \(h(x)\) is an irreducible factor of \(g(x)\), then the
degree of \(f\) divides the degree of \(h\).

Topic Tags

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