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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2008 • Question 3

Question

Math 6310 • Algebra Fall 2008 • Question 3

Let \(K\) be a field and let \(f(x)\in K[x]\) be a monic polynomial of degree \(n\) with \(f(0)\ne 0\). Suppose \(f(x)\) has \(n\) distinct roots in its splitting field, and that the set of roots is closed under multiplication. Determine \(f(x)\).

Raw LaTeX

Let \(K\) be a field and let \(f(x)\in K[x]\) be a monic polynomial of degree \(n\) with
\(f(0)\ne 0\). Suppose \(f(x)\) has \(n\) distinct roots in its splitting field, and that the set of
roots is closed under multiplication. Determine \(f(x)\).

Topic Tags

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