[Math] Is $F[x,y]$ a Euclidean Domain

Question

I was wondering if this is just common knowledge. So far for a field $F$ and transcendental $x$ and $y$, I know one can define the degree by

$1) \deg c =0$, for any $c \in F-\{0\}$

$2) \deg x^{n_1}y^{n_2}=n_1+n_2$, for any $n_1, n_2 \in \mathbb{N}$

$3)\deg (f+g)=\max\{\deg f , \deg g\}$, for any $f, g \in F[x,y]$

but I have yet to prove this is a Euclidean measure on $F[x,y]$, so my suspicion is that it is not. I've also been having a hard time finding this particular problem online. If anyone can either solve this or point me to an online resource with the answer, it would be greatly appreciated.

In general, I'm interested in knowing whether $F[x_1, x_2, \dots ,x_k]$ is a Euclidean Domain.

Answer 1

Every Euclidean domain is a principal ideal domain, but the ideal $(x,y)$ in $F[x,y]$ is not principal.