# Kernel of a field homomorphism

1. Apr 6, 2005

### ti89fr33k

Show that the kernel of a field homomorphism is either the trivial homomorphism or isomorphic to the field.

I've tried to see it as a factor group, but I'm stuck. Can someone help?

mary

2. Apr 6, 2005

### matt grime

I'm sorry, are you think ing of kernels as morphisms or objects? Cos you refer to the kernel as a homomorphism and some thing that is isomorphic to the field.

In any case the kernel of a ring homomorphism (of which a field homomrphism is a special case) is an ideal. So how many ideals of a field are there?

Last edited: Apr 6, 2005