Addition table for a finit field with 3 elements $\mathbb{F}_{3}=\{0,1,-1\}$

Question

just a quick question :-)

I need to make an addition table for a finit field with 3 elements $\mathbb{F}_{3}=\{0,1,-1\}$

Is this table correct?

$ \begin{matrix} \boldsymbol{\textbf{+}} & \mathbf{0} & \textbf{1} & \textbf{-1} \\ \boldsymbol{\textbf{0}} & 0 & 1 & -1\\ \boldsymbol{\textbf{1}} & 1 & 0 & 1 \\ \boldsymbol{\textbf{-1}} & -1 & 1 & 0 \\ \end{matrix}$

@ZeroTheHero:

$ \begin{matrix} \boldsymbol{\textbf{+}} & \mathbf{0} & \textbf{1} & \textbf{2} \\ \boldsymbol{\textbf{0}} & 0 & 1 & 2 \\ \boldsymbol{\textbf{1}} & 1 & 2 & 0 \\ \boldsymbol{\textbf{2}} & 2 & 0 & 1 \\ \end{matrix}$

Answer 1

You can't say that 1+(-1) = 0 = 1 So this is wrong because this 'field' now only has 2 elements, and also, a field requires that 1!=0. The alternative is that the -1 is an abuse of notation is not referring to additive inverse of 1, in which case this field is stupid lol. Theres only one possible way to define F_3 I'm pretty sure. Show That There Is One and (Essentially) Only One Field With 3 Elements