If F is a Field it is known that a polynomial f(x) of degree 2 or 3 in F[x] is irreducible if and only if f(x) has no zeros in F. Show that this is not necessarily true for degree 4 (hint: in R[x] find a polynomial with no zeros, that is the square of a polynomial of degree 2)

If F is a Field it is known that a polynomial f(x) of degree 2 or 3 in F[x] is irreducible if and only if f(x) has no zeros in F. Show that this is not necessarily true for degree 4 (hint: in R[x] find a polynomial with no zeros, that is the square of a polynomial of degree 2)

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 18E: Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily...

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Question

If F is a Field it is known that a polynomial f(x) of degree 2 or 3 in F[x] is irreducible
if and only if f(x) has no zeros in F. Show that this is not necessarily true for degree
4 (hint: in R[x] find a polynomial with no zeros, that is the square of a polynomial of
degree 2)

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