Proving F is Finite: A Perspective on Ring Homomorphisms

[dogma]

Let $$f: Z \rightarrow F$$ be a ring homomorphism from Z onto a field F. Prove that F must be finite with a prime number of elements.

How would one go about proving this? I understand that multiplication and addition must be preserved in a homomorphism. I guess I must somehow show that a proper factor ring of Z is finite, but I'm not sure how.

I'd greatly appreciate any help. Thanks!

 

[Euclid]

Look at the kernel. Suppose it's {0}. Then F is infinite, isomorphic to Z. Z isn't a field, so that's no good. So the kernel is nZ for some nonzero n. So F is isomorphic to Z/nZ for some n.

 

[dogma]

Thanks, Euclid.

That was the ticket to get me on the right path.

Take care.

 

[mathwonk]

another point of view is to recognize that an onto homomorphism is another way of thinking of a qupotient construction in the opriginal ring.

i.e. up to isomorphism, the only possible onto homomorphisms are of form R-->R/I, where I is an ideal in R. So just ask what the ideals are in Z. The only ones that give f8ields are maximal ideals, and the only maximal ideals in Z are of form Zp where p is prime, so the only possible fields of form Z/I are the finite fields Z/p.

this is another view on the same answer above. but the moral is that all the information about an onto homomorphism is already contained in the original ring. i.e. an onto map is just a way of making identifications in the original ring.