Ideals in a field and relation to homomorphisms.

Question

For a quiz coming up, my professor has asked the class to

"know what the ideals in a field look like, and what this tells you about homomorphisms from a field to other rings..."

I have looked in the textbook but their description is confusing to me and I'm having trouble understanding the concept. Can anyone explain this in a simple way or perhaps link me to a resource that can?

Thanks.

Answer 1

Hint: Consider $a\in F$ and the ideal generated by $a$. Consider the case $a=0$ and $a\ne0$.

Answer 2

Let ideal $I$ of field $F$ contain some $a\neq0$.

In a field every element $\neq0$ is a unit, so for an arbitrary $b\in F$ we have $b=ba^{-1}a$.

When it comes to $I$ then what can be concluded about $b$ on base of this?

Answer 3

I think what he means is this, the kernel $\ker \varphi = \{x\in\mathbb{F}: \varphi(x)=0\}$ in a field is always an ideal and as such you can know something about the possible homomorphisms if you know the ideals of the ring, or in our case, field.

However in a field you have only 2 ideals, namely the trivial ideals which is the 0 ideal and all of $\mathbb{F}$. This is because all non-zero elements are units and as such we have that if $a\in I$ and $a\neq 0$ then $$b=ba^{-1}a\in I$$ which means you have either a homomorphism that only sends $0\to 0$ or $\mathbb{F}\to 0$ as the only possible options.