I need some help proving the following:
Let $\mathbb{F} = \{0, 1, a, b\}$ be a field with four elements. Prove that $a^2 = b$.
You can use $a \cdot 0 = 0$ without proving it.
Attempted solution:
$$ a^2 = b \\ aa = b \\ aa + 0 = b + 0 $$ We know $a \cdot 0 = 0$ so we can substitute for zero $$ aa + a \cdot 0 = b $$ Using the additive inverse of $aa$ we get: $$ (aa) + (-aa) + a \cdot 0 = b + (-aa) \\ a \cdot 0 = b + (-aa) $$ I’m still not getting the concept of how to prove things, maybe a little insight into what are possible steps to approach problems like these.
Thats as far as I got, any help is appreciated.