Abstract Algebra: Isomorphic polynomial rings

1. Dec 2, 2008

katarinab316

1. The problem statement, all variables and given/known data

If F is an infinate field, prove that the polynomial ring F[x] is isomorphic to the ring T of all polynomial functions from F to F

2. Relevant equations

3. The attempt at a solution

T is isomorphic to F[x]
f(a+b) = f(a) + f(b)
f(ab)=f(a)f(b)
It is surjective by the definition of T
Injective: If f is not equal to g in F[x], then f(x) is not equal to g(x) in T.
Let h=f-g and assume h is not 0 in F
Claim: h(x) is not equal to zero in T, then h(a)=0 for every a in F. Thus every a in F is a root of the polynomial h in F[x}. However, there is no nonzero polynomial with infinately many roots, thus h=0 in F, which contradicts the asumption of h is not 0 in F, so injective. So F[x] is isomorphic to the ring T.

2. Dec 2, 2008

Hurkyl

Staff Emeritus
Hrm. You haven't said what you want from us. I assume just to review your work?

This doesn't make sense. Why are you starting with the thing you're trying to prove? What is f? a? b? What was the point of those equalities? Why are they true? What is surjective? (I assume some of these latter questions, as well as what follows, would become clearer after you answer the first few)

3. Dec 3, 2008

katarinab316

Well basically this is an assignment turned in that got a "redo". As you can probably tell, I'm so confused in this subject I don't even know how to begin to correct it....thanks for the input so far