Polynomial ring over a field is uniquely Euclidean with norm equal to degree

Statement
Suppose $$k$$ is a field, and $$k[x]$$ is the polynomial ring in one variable over $$k$$. Then, $$k[x]$$ is a uniquely Euclidean domain. Specifically, the function that sends a nonzero polynomial to its degree is a uniquely Euclidean norm on $$k[x]$$.

Additional facts
Here are some further facts about this norm:


 * The degree is a filtrative norm: The set of all polynomials of degree less than $$n$$, along with the zero polynomial, is an additive subgroup of the polynomial ring. In particular, the degrees of the sum and difference of two polynomials is bounded from above by the maximum of their degrees.
 * The degree is a multiplication-additive norm: The product of two polynomials has degree bounded by the sum of their degrees.
 * The degree is an automorphism-invariant norm: This follows from the fact that the automorphisms of the polynomial ring in one variable are precisely the maps that send $$x$$ to $$x - a$$ for some field element. In particular, these maps preserve the degree.