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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 1999 • Question 4

Question

Math 6310 • Algebra Fall 1999 • Question 4

Let \(F\) be a field, and let \(f(X)\) be a polynomial in \(F[X]\). Show that \(f(X)\) has a multiple root in some extension of \(F\) if and only if \(f(X)\) and its derivative \(f'(X)\) are not relatively prime.

Raw LaTeX

Let \(F\) be a field, and let \(f(X)\) be a polynomial in \(F[X]\). Show that \(f(X)\) has a multiple
root in some extension of \(F\) if and only if \(f(X)\) and its derivative \(f'(X)\) are not
relatively prime.

Topic Tags

Ext Functors and Module Extensions Field Extensions Irreducible Polynomials Polynomial Rings and Prime Ideals