Let f(x) and g(x) be irreducible polynomials over a field F and let a and b belong to some extension E of F. If a is a zero of f(x) and b is a zero of g(x), show that f (x) is irreducible over F(b) if and only if g(x) is irreducible over F(a)

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.3: Factorization In F [x]
Problem 8E: Let be an irreducible polynomial over a field . Prove that is irreducible over for all nonzero ...
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Let f(x) and g(x) be irreducible polynomials over a field F and let a and b belong to some extension E of F. If a is a zero of f(x) and b is a zero of g(x), show that f (x) is irreducible over F(b) if and only if g(x) is irreducible over F(a)

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