When $0$ is mentioned at anytime when talking about Fields, does this mean we are talking about the number $0$, or is it the additive identity?

Question

When talking about fields, such as the field axioms and the theorems that follow, when $0$ is mentioned at anytime, does this mean we are talking about the number $0$, or is it the additive identity?

Answer 1

I assume that by the number zero you are referring to $0$ as an element of $\mathbb{N}$.

Say we have a field $\mathbb{F}$ and denote the zero element of $\mathbb{F}$ by $0_{\mathbb{F}}$, then statements like $$ \alpha=0,\alpha\cdot0,\alpha+0 $$

are referring to $0_{\mathbb{F}}$. However statements like $$ |\{a\in\mathbb{F}\mid\text{a satisfies...}\}|=0 $$

are referring to the number zero, that is, as a counting number.

I hope that the difference is clear from the examples.