Prove that $\langle\sqrt2\rangle$ is a maximal ideal in $\Bbb Z[\sqrt2]$. How many elements are in $\Bbb Z[\sqrt2]/\langle\sqrt2\rangle$?

Question

Prove that $\langle\sqrt{2}\rangle$ is a maximal ideal in $\Bbb Z[\sqrt{2}]$. How many elements are in the ring $\Bbb Z[\sqrt{2}]/\langle\sqrt{2}\rangle$ ?

I am unable to solve this. Please help me on this.

Answer 1

Define

$$\phi:\Bbb Z[\sqrt2]\to\Bbb Z/2\Bbb Z\;,\;\;\phi(a+b\sqrt2):=a\mod2$$

Prove this is ring homomorphism , find its kernel and apply the first isomorphism theorem.