Let F be a field and $G=F\times F$ Define addition by $(a,b)+(c,d)=(a+c,b+d)$ and multiplication by $(a,b)\cdot(c,d)=(ac,bd)$
Does these operations define a field on G?
I'm fairly comfortable with the addition part, however its the multiplication part that trips me up. Surely $(ac)^{-1}$ exists since F is a field, and $\frac{1}{a}, \frac{1}{c}$ are each in F since we assumed that F is a field.