Question

Show that every homomorphic image of a field $F$is isomorphic either to$F$ itself or to the zero ring

Short answer

It can be proved that every homomorphic image of a field$F$ is isomorphic either to$F$ itself or to the zero ring.

Step-by-step solution

Results used

In reference to Exercise 10 in section 6.1,

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

Show kernel is an ideal 

Let $R$be a homomorphic image of $F$.

That is $\theta :F\to R$ is a surjective homomorphism.

Then the kernel is an ideal of $F$

By the above-mentioned result,

kernel is either $\left(0\right)$or $F$.

If the kernel is 0

If the kernel is $\left(0\right)$, then $\theta$is injective.

Therefore, $\theta$is isomorphism.

Hence, $F$is isomorphic to itself.

If the kernel is F

If the kernel is $F$, then $\theta$carries every element to $\left(0_R\right)$.

Hence, $F$is isomorphic to the zero ring.

Conclusion

$F$is isomorphic either to$F$ itself or to the zero ring.