Show that $\mathbb Z_2$ is the only field with two elements

Question

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In a field there is the additional requirement that $0\ne1$, hence a field has at least two elements. Show that $\mathbb Z_2$ is the only field with two elements.

Answer 1

Let $F$ be a field with only two elements {$0',1'$}.

Note that for addition table we have $$ 0'+0'=0', 0'+1'=1', 1'+1'=0',1'+0'=1'$$

For multiplication we have $$ 0'*0'=0',0'*1'=0',1'*0'=0', 1'*1'=1'$$

That defines an isomorphism $$ \phi : F\to \mathbb Z_2 $$ between our field $F$ and $ \mathbb Z_2$ defined by $$\phi (0')=0\text { and } \phi (1')=1$$