Gauss's Lemma

Let be a Unique Factorisation Domain with a Field of Fractions .
Suppose is primitive and irreducible in .
Then is irreducible in .

Lemma

1. A prime element in is also prime in .
2. If are primitive then is also primitive
3. If , then up to associates

Lemma

Let and is primitive.
If in , then in , where is the Field of Fractions of .