How to show every field is a Euclidean Domain.
You're probably overthinking it. Let $q = x \cdot y^{-1}$, which always exists, because $F$ is a field, and $y \ne 0$. So let $r$ be zero, and you don't need to worry at all about the valuation.
How to show every field is a Euclidean Domain.
You're probably overthinking it. Let $q = x \cdot y^{-1}$, which always exists, because $F$ is a field, and $y \ne 0$. So let $r$ be zero, and you don't need to worry at all about the valuation.