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?
No! Why would J = R if J contains r + si = j??? J = R iff 1 is in J, and this is what you must show.
2. To show that
is a field we know this from the fact that
is maximal. How do you show it has 9 elements?