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).
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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![34. 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).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F174d7d89-86f6-4fd3-817b-704de3f40cca%2Fd85a9e91-f74a-4750-a958-13731c8a55d0%2Fvrrjkp_processed.png&w=3840&q=75)
Transcribed Image Text:34. 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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