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Abstract Algebra: An Introduction

Found in: Page 149

Book edition 3rd

Author(s) Thomas W Hungerford, David Leep

Pages 608 pages

ISBN 9781111569624

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Short Answer

If $I$ is an ideal in a field $F$ , prove that $I = (0_{F})$ or $I = F$ .

It can be proved $I = (0_{F})$ or $I = F$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Previous results used

We shall use the following result to prove this.

Given $R$ is a ring with identity and $I$ is an ideal in $R$ .

If $1_{R} \in I$ , then $I = R$ .

Proving the first condition

If $I = (0_{F})$ , there is nothing to prove.

Proving the second condition

If $I \neq (0_{F})$ , there exists $a \in I$ such that $a \neq 0_{F}$ .

Now, we know that $F$ is a field.

Therefore, $a$ must be a unit.

Hence, from the above result, we will conclude that $I = F$ .

Conclusion

Hence, $I = (0_{F})$ or $I = F$ .

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