Theorem. Let F be a field and ƒ € F[x] a polynomial of degree n. Then there is a finite-dimensional extension of F in which f factors into linear factors f(x) = (x – a1) · … · (x – a,n). Proof. Apply the last theorem repeatedly to get getting extensions of extensions and factor out a linear factor each time until the degree is reduced to 1.

Theorem. Let F be a field and ƒ € F[x] a polynomial of degree n. Then there is a finite-dimensional extension of F in which f factors into linear factors f(x) = (x – a1) · … · (x – a,n). Proof. Apply the last theorem repeatedly to get getting extensions of extensions and factor out a linear factor each time until the degree is reduced to 1.

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 6TFE

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