A field with four elements

Question

Determine the additive group of the field of four elements.

My attempt:Consider $(F,+,.) $ the field of four elements.Now $0,1\in F$ as $(F,+,.) $ is a field .As it contains $4$ elements $\exists a\neq 0,1\in F$.Corresponding to $a $ we should have $-a$.This is all I have got.Is it correct?Please suggest edits if required

Answer 1

This field should have characteristic $2$, which means $\alpha+\alpha=0$. That is $-\alpha=\alpha$. You might try $1+\alpha$ as the fourth element.