We are given that $a \oplus b = a+b+1$ and $a \odot b= a+b+ab$ and I need to prove that it is a field. Do I just need to know that there is an inverse for $a \odot b$? So if I could let $a+b+ab=c$ then I want to find a $-c$ such that $c \odot -c=1$? Then the inverse, $-c=\frac{1}{a+b+ab}$ then? This would be that for every c I have a multiplicative inverse and since the set is a infinite I will not have any zero divisors and I have an integral domain. And since addition and multiplication is commutative here I also have a field.

Am I on the right track?