- #1

ehrenfest

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**[SOLVED] fundamental theorem of algebra**

## Homework Statement

Theorem: If F is a field, then every ideal of F[x] is principal.

Use the above theorem to prove the equivalence between these two theorems:

**Fundamental Theorem of Algebra:**Every nonconstant polynomial in C[x] has a zero in C.

**Nullstellansatz for C[x]:**Let f_1(x), ...,f_r(x) in C[x] and suppose the every alpha in C that is a zero of all r of these polynomials is also a zero of a polynomial g(x) in C[x]. Then some power of g(x) is in the smallest ideal of C[x] that contains the r polynomials f_1(x),..., f_r(x).

## Homework Equations

## The Attempt at a Solution

Lets prove the FTA implies that Nullstellansatz first. Say p(x) generates the ideal. I think its pretty clear that all the alpha's will be zeroes of p(x), right? So, what does that mean about the relationship between p(x) and powers of g(x). All the alphas are zeroes of both...but why does that necessarily mean p(x)q(x)=g(x)^n for some q(x)...