1. Prove that is a subring of and that is a maximal ideal in .
Showing it's a subring is obvious. Just show closure under addition and multiplication and that . Also show that the solution of the equation is in .
To show that is maximal consider the following: means that there is some such that where . So . And any ideal that contains must contain so that .
Is this correct?
2. To show that is a field we know this from the fact that is maximal. How do you show it has 9 elements?