# Math Help - Factor Rings of Polynomials Over a Field

1. ## Factor Rings of Polynomials Over a Field

On page 222 of Nicholson: Introduction to Abstract Algebra in his section of Factor Rings of Polynomials Over a Field we find Theorem 1 stated as follows: (see attached)

Theorem 1. Let F be a field and let $A \ne 0$ be an ideal of F[x]. Then a uniquely determined monic polynomial h exists exists in F[x] such that A = (h).

The beginning of the proof reads as follows:

Proof: Because $A \ne 0$, it contains non-zero polynomials and hence contains monic polynomials (being an ideal) ... ... etc. etc.

BUT! why must A contain monic polynomials??

Help with this matter would be appreciated!

Peter

2. ## Re: Factor Rings of Polynomials Over a Field

Hi,
Suppose $g(x)=a_nx^n+... \text{ is in A with }a_n\ne0$.
Since A is an ideal, $a_n^{-1}g(x)}$ is monic and is in A.