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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2016 • Question 10

Question

Math 6310 • Algebra Fall 2016 • Question 10

Let \(L/K\) be a Galois extension of fields of characteristic zero. Suppose \(f(x)\) is an irreducible polynomial of degree 5 in \(K[x]\). If \(f(x)\) has no roots in \(L\), prove that it is irreducible in \(L[x]\).

Raw LaTeX

Let \(L/K\) be a Galois extension of fields of characteristic zero. Suppose \(f(x)\) is an
irreducible polynomial of degree 5 in \(K[x]\). If \(f(x)\) has no roots in \(L\), prove that it is
irreducible in \(L[x]\).

Topic Tags

Field Extensions Galois Groups of Polynomials Galois Theory Irreducible Polynomials