I was wondering if someone could give an opinion on some work as I struggle to know when I'm on the right track or not with groups.
So for: $ (\mathbb R,\circ)$ where $ x \circ y := (x+y)^2$
Is there a neutral element?
At first I wanted to solve $ (x \circ e )^2 $ and show there was no possible solution but I wasn't sure if that was a correct way to do it or if I would even know it is not a possible solution for all $x$.
However, I was wondering if now would it be sufficient to use $ x = -1 $ as a counterexample so that it can be seen:
$[(-1)+e]^2=-1$
Which clearly cannot be true as no value within $\mathbb R$ raised to the power of $2$ can give a minus value.