Multiplicative group of a field (not necessarily finite)

Question

Let $F$ be any field, is $F^{\times}=F\setminus\{0\}$ always true? I know this is true for finite fields. What about infinite fields?

Answer 1

This is always true, typically by definition. It is common, for fields $F$, to define $$ F^\times := F \setminus \{0\} $$ More broadly, for rings $R$, one can define $$\begin{align*} R^\times &:= \{ r \in R \mid \text{$r$ a unit in $R$} \} \\ R_{\ne 0} &:= R \setminus \{0\} \end{align*}$$ These are generally not equal, but if $R$ is a field, then $R_{\ne 0} = R^\times$. That is to say, $R^\times$ is the set of invertible elements, and when $R$ is a field, then all nonzero elements are invertible.