Question

If$I$ is an ideal in a field$F$ , prove that$I=\left(0_F\right)$ or$I=F$ .

Short answer

It can be proved$I=\left(0_F\right)$ or$I=F$ .

Step-by-step solution

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 $I=\left(0_F\right)$, there is nothing to prove.

Proving the second condition

If $I\ne \left(0_F\right)$, there exists$a\in I$ such that$a\ne 0_F$ .

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,$I=\left(0_F\right)$ or$I=F$ .