# Unique factorization domain, roots of a polynomial, abstract algebra

1. Sep 9, 2012

### rayman123

1. The problem statement, all variables and given/known data
let A be a UFD and K its field of fractions. and $$f\in A[x]$$ where $$f(x)=x^{n}+a_{n-1}x^{n-1}+....+a_{1}x+a_{0}$$ is a monic polynomial. Prove that if f has a root $$\alpha=\frac{c}{d}\in K$$,$$K=Frac(A)$$ then in fact $$\alpha\in A$$

I need some guidance with the proof.
Proof:
$$f(\alpha)=0\Rightarrow c^{n}+a_{n-1}c^{n-1}d+...+a_{1}cd^{n-1}+a_{0}d^{n}=0$$
which gives
$$c^{n}=-a_{n-1}c^{n-1}d-...-a_{1}cd^{n-1}-a_{0}d^{n}$$
and we observe that $$d|c^{n}$$*since all the terms on the rhs are multiples of d
but how do we know that $$c^{n}|d$$??? from what do we conclude that
And later in the proof it says '' hence c,d are relatively prime we get that $$\frac{c}{d}\in A$$ how do we deduce the last part as well?
Any help appreciated!

1. The problem statement, all variables and given/known data
how do we know that $$c^{n}|d$$

2. The problem statement, all variables and given/known data
how do we know that c, d are relatively prime?

3. The problem statement, all variables and given/known data
based on what do we conclude that $$\frac{c}{d}\in A$$