I have just proved $R=\mathbb{Z}[\sqrt{2}]$ is a UFD.
Now I am trying to factorise $2, 3,$ and $7$ into irreducibles in $R$, but I'm not sure how to go about this.
I know $7=(3+\sqrt{2})(3-\sqrt{2})$ and $2=\sqrt{2}\sqrt{2}$ so think I am done factorising these, but don't know how to prove these factors are irreducible. I also feel I found the factors of 7 more through luck than anything else, so wonder if there is a more systematic way to factorise.