Let $R$={$a+b\sqrt{2}$ | a, b integers} and $M$={$a+b\sqrt{2}$ | 5|$a$ and 5|$b$} Prove $M$ is a maximal ideal of $R$ and $\frac{R}{M}$ is a field having 25 elements
I could prove that $M$ is maximal ideal of $R$ and $\frac{R}{M}$ is a field, but I can't prove that field has 25 elements
Idea:
For any $a+b\sqrt{2}$, where $a,b \in \Bbb{Z}$, observe that we can have $a \equiv r \pmod{5}$ and $b \equiv s \pmod{5}$ with $r,s \in \{0,1,2,3,4\}$. Thus in the quotient ring $$(a+b\sqrt{2})M =(r+s\sqrt{2})M.$$ Since every element $(a,b)$ can be identified with a corresponding $(r,s)$, thus the number of distinct representatives is $25$.