# Thread: Polynomial Rings - Gauss's Lemma

1. ## Polynomial Rings - Gauss's Lemma

I am trying to understand the proof of Gauss's Lemma as given in Dummit and Foote Section 9.3 pages 303-304 (see attached)

On page 304, part way through the proof, D&F write:

"Assume d is not a unit (in R) and write d as a product of irreducibles in R, say $\displaystyle d = p_1p_2 ... p_n$ . Since $\displaystyle p_1$ is irreducible in R, the ideal $\displaystyle (p_1)$ is prime (cf Proposition 12, Section 8.3 - see attached) so by Proposition 2 above (see attached) the ideal $\displaystyle p_1R[x]$ is prime in R[x] and $\displaystyle (R/p_1R)[x]$ is an integral domain. ..."

My problems with the D&F statement above are as follows:

(1) I cannot see why the ideal $\displaystyle (p_1)$ is a prime ideal. Certainly Proposition 12 states that "In a UFD a non-zero element is prime if and only if it is irreducible" so this means $\displaystyle p_1$ is prime since we were given that it was irreducible. But does that make the principal ideal $\displaystyle (p_1)$ a prime ideal? I am not sure! Can anyone show rigorously that $\displaystyle (p_1)$ a prime ideal?

(2) Despite reading Proposition 12 in Section 8.3 I cannot see why the ideal $\displaystyle p_1R[x]$ is prime in R[x] and $\displaystyle (R/p_1R)[x]$ is an integral domain. ...". (Indeed, I am unsure that $\displaystyle p_1R[x]$ is an ideal!) Can anyone show explicitly and rigorously why this is true?

I would really appreciate clarification of the above matters.

Peter

2. ## Re: Polynomial Rings - Gauss's Lemma

In trying to answer my problem (1) above - I cannot see why the ideal $\displaystyle (p_1)$ is a prime ideal - I was looking at definitions of prime ideals and trying to reason from there.

I just looked up the definition of a prime element in D&F to find the following on page 284:

The non-zero element $\displaystyle p \in R$ is called prime if the ideal (p) generated by p is a prime ideal!

So the answer to my question seems obvious:

$\displaystyle p_1 irreducible \Longrightarrow p_1$ prime $\displaystyle \Longrightarrow (p_1)$ prime ideal

Although this now seems obvious, I would like someone to confirm my reasoning (which as I said now seems blindingly obvious! :-)

Peter

3. ## Re: Polynomial Rings - Gauss's Lemma

If we have $\displaystyle f \in K[x]$ for some polynomial ring $\displaystyle K[x]$ then the following are equivalent
1) f is irreducible
2) (f) is prime
3) (f) is maximal