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.

For any $x\in (F,+,\cdot)$ we require that if $x\neq 0_F$ then there exists an element $x^{-1}$ so that $x\cdot x^{-1}=x^{-1}\cdot x=1_F$. Consider $(r,0_S)\in R\times S$. Supposing that there is an element $(r,0_S)^{-1}=(a,b)\in R\times S$ then we would have $(r,0_S)\cdot(a,b)=(1_R,1_S)=(1,1)_{R\times S}$ (it is easy to check the last equality). This would force $0_S\cdot b=1_S$. This is a contradiction since $S$ is a field and we must have $s\cdot 0_S=0_S$ for all $s\in S$. So $(r,0)$ has no inverse and $R\times S$ cannot be a field.
