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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2012 • Question 8

Question

Math 6310 • Algebra Fall 2012 • Question 8

Let \(K\subseteq L\) be fields, and let \(f(x)\) be an irreducible polynomial in \(K[x]\). If there exists \(a\) in \(L\) with \(f(a)=0=f(a^2)\), prove that \(f(x)\) splits in \(L[x]\).

Raw LaTeX

Let \(K\subseteq L\) be fields, and let \(f(x)\) be an irreducible polynomial in \(K[x]\). If there
exists \(a\) in \(L\) with \(f(a)=0=f(a^2)\), prove that \(f(x)\) splits in \(L[x]\).

Topic Tags

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