I have the following problem and I'm stuck in the second part:
Let $K$ be a field. We define in $K \times K$ the next operations: $$(a, b) + (c, d) := (a + c, b + d)$$ $$(a, b) · (c, d) := (ac − bd, ad + bc)$$
- Prove that $K\times K$ is a commutative and unitary ring with the given operations. Done
- Prove that $K \times K$ is a field iff $K$ satisfies that for all $a,b\in K$ that $a^2 + b^2 =0$ then $a=b=0$.
Any hints? Thanks!