I'm currently working through an abstract algebra text book on my own, and I'm stuck on this problem that has no solution given:
Let $K$ be a field, and $0 \neq a \in K$. We define $L := K^2$ with the following set of operations:
$(x,y) + (x^\prime, y^\prime) := (x + x^\prime, y + y^\prime)$
$(x,y) * (x^\prime, y^\prime) := (xx^\prime + yy^\prime a, xy^\prime + x^\prime y)$
This defines a commutative ring structure with one over $L$, with $0_L = (0,0)$ and $1_L = (1,0)$.
Please show that the following holds true:
$L$ is a field $\Leftrightarrow$ $\not\exists b\in K : b^2 = a $
How do I approach this? I assume I have to check the properties that cause a Ring $R$ to become a full field, but I'm just stumped here.