Let F be a field, let k be a non-negative integer, and define a function $d$ from the set F* of non-zero elements of F to the set of non-negative integers by $d(a)=k \quad \forall a\in F^*$. Prove that $d$ is a Euclidean function on F and that F is a Euclidean domain.
I know that the definition of a Euclidean function is
Let R be an integral domain. A Euclidean function on R means a function $d:R^*\mapsto \mathbb{N}\cup {0}$,
$R^*={\{r\in R| r \neq0}\}$ such that
i.) $d(ab)\ge d(a) \quad \forall a,b\in R^*$;
ii.) If $a,b\in R$ with $b \neq0$ then there exists $q,r\in R$ such that $a=bq+r$ and either $r=0$ or $d(r) \lt d(b)$.
and a Euclidean domain is an integral domain for which there exists a Euclidean function.
But I am unsure as how to apply this definition.
You can divide the problem in two parts. First prove that a field is an integral domain, and then prove that the function $d$ is an Euclidean function.
For the first part, recall the definition of a field and the definition of an integral domain. Mainly that a field is commutative, and that all non-zero elements are invertible.
For the second part, you can check that the first condition is trivially satisfied. For the second condition, try $q=b^{-1}a$ (why can we do this?)