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?
It is clear that every odd prime in has a form either 1(mod 4) or 3(mod 4) and an even prime 2=(1+i)(1-i) is reducible in . If , then with x and y integers by Fermat's two squares theorem. Further, and neither nor is a unit because their norms are bigger than 1. Thus is reducible.
Since is an integral domain (not a field) and a PID, we have as irreducible elements and are maximal ideals in (For example, <3>, <7>, <11>, <19>, ..., are maximal ideals in ).