What is the kernel of a field morphism and how is it related to ideals?

[eep]

Does it make sense to talk about the kernel of a field morphism? If so, what is it? I'm getting confused because we've defined a field to be a commutative group (F,+) and a map m: F -> F s.t. (F \{0}, m) form another commutative group. For shorthand we're calling the unit element for the + operation 0, and the unit element for m as 1.

So I'd want to define the kernel of a field morphism as the set of all elements in F1 that get mapped to 0 in F2, and the set of all element in F1 that get mapped to 1 in F2. Help!

 

[ircdan]

Does it make sense to talk about the kernel of a field morphism? If so, what is it? I'm getting confused because we've defined a field to be a commutative group (F,+) and a map m: F -> F s.t. (F \{0}, m) form another commutative group. For shorthand we're calling the unit element for the + operation 0, and the unit element for m as 1.

So I'd want to define the kernel of a field morphism as the set of all elements in F1 that get mapped to 0 in F2, and the set of all element in F1 that get mapped to 1 in F2. Help!

the kernel of a group hom f:G->G is just kerf = {x in G| f(x) = e}
the kernel of a ring hom f:R->R is just kerf = {x in R| f(x) = 0}

The definitions are analogous. They are both the set of elements x such that f(x) = identity in the group

A field is a ring, with some other nice properties, so just set R = F above.

 

[eep]

I was getting confused because in a field we have two groups, (F, +) and (F, .). Thanks.

 

[morphism]

I was getting confused because in a field we have two groups, (F, +) and (F, .). Thanks.

(F, .) is not a group; (F\{0}, .) is.

Also, when you say "field morphism" you're probably talking about a ring (homo)morphism (after all, a field is a special type of ring), which already has a notion of a kernel. If you stop to think about what a kernel ought to do (think in the flavor of injectivity, equivalence relations, quotient structures, etc.), you should see that considering things that get mapped to 1 isn't really what we're after.

 

[Kummer]

As morphism says field-homomorphism can be though of as ring-morphisms. However, the kernel of a field-homomorphism is not very interesting. Because by the fundamental theorem of homomorphisms its kernel is an ideal of a field. But fields only have the identity or themselves as their own ideals. Thus, it is very trivial to consider.

 

[eep]

What book would be good for these sorts of notions? I'm currently taking a linear algebra class but the professor felt it was best to first talk about mathematical objects. So we defined monoids, groups, rings, fields, vector spaces, and algebras. We then spoke about morphisms between two of the same object, so I'm not familiar with terms like equivalence relations and quotient structures.

A homework question asked us to show that a field morphism has to be injective, which led me to my question. It is trivial to show now that I know how to think about a field morphism!

 

[morphism]

Any book on abstract algebra would go over these things. There are a few threads about algebra books scattered across the forums, so try searching for them.

 

[mathwonk]

a field morphism is a ring morphism, so the kernel of a field morphism is an ideal. so ity suffices to classify ideals in a field. ...?