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Abstract Algebra: An Introduction
Found in: Page 94
Abstract Algebra: An Introduction

Abstract Algebra: An Introduction

Book edition 3rd
Author(s) Thomas W Hungerford, David Leep
Pages 608 pages
ISBN 9781111569624

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Short Answer

If F is a field, show that F[x] is not a field.

It is proved that F[x] is not a field.

See the step by step solution

Step by Step Solution

Step 1: Polynomial Arithmetic

If any given function R[x] is a ring, then the commutative, associative, and distributive laws hold such that the function f(x)+g(x) exists.

Step 2: Fields:

It is given that F is a field.

Let us assume that F[x] is also a field. Then, xFx will have an inverse as:


Therefore, in this case, we have:


Here, the constant coefficient is zero.

According to the theorem 4.1, it should be 1.

This shows our assumption is invalid.

Hence proved, F[x] is not a field.

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