If $R$ and $S$ are fields, either prove or disprove that $R\times S$ is a field That's the question from my homework. I am thinking $R\times S$ is not a field, but I'm not sure. I understand the definition of a field, but I am not sure how to proceed.

If $R^*$ is the set of units of a ring, then $(R \times S)^* = R^* \times S^*$. A commutative ring $R$ with unit is a field if and only if $R^* = R - \{0\}$. If $R \times S$ is a field, we would have the following equality. $$(R \times S)^* = R^* \times S^* = (R \times S) - \{(0,0)\}$$
Is this possible?
