Is 11 a Prime in the Non-UFD Ring Z[sqrt{-5}]?

[bessletama]

I am trying to prove that $11$ is a prime in $\mathbb{Z}[\sqrt{-5}]$.

I noticed that $\mathbb{Z}[\sqrt{-5}]$ is not a UFD so I cannot show that it is irreducible then conclude it is prime.

I know that that an ideal is prime if and only if the quotient ring is a domain.
I was wondering if it is correct for me to show that
$\mathbb{Z}[\sqrt{-5}]/(11)\cong\mathbb{Z}_{11}[x]/(x^2+1)$
If this is true then I can conclude that $\mathbb{Z}[\sqrt{-5}]$ is a domain because
$\mathbb{Z}_{11}[x]/(x^2+1)$ is a finite field.
Thank you

EDIT: OMG, Made a huge typo originally. The ring is $\mathbb{Z}[\sqrt{-5}]$ not $\mathbb{Z}[\sqrt{5}]$

 

[DonAntonio]

I am trying to prove that $11$ is a prime in $\mathbb{Z}[\sqrt{5}]$.

I noticed that $\mathbb{Z}[\sqrt{5}]$ is not a UFD so I cannot show that it is irreducible then conclude it is prime.

I know that that an ideal is prime if and only if the quotient ring is a domain.
I was wondering if it is correct for me to show that
$\mathbb{Z}[\sqrt{5}]/(11)\cong\mathbb{Z}_{11}[x]/(x^2+1)$
If this is true then I can conclude that $\mathbb{Z}[\sqrt{5}]$ is a domain because
$\mathbb{Z}_{11}[x]/(x^2+1)$ is a finite field.
Thank you

You could conclude that if you can show the isomorphism $\,\mathbb{Z}[\sqrt{5}]/(11)\cong\mathbb{Z}_{11}[x]/(x^2+1)\,$ .

In fact, you'd conclude something stronger: the ideal $\,(11)\subset \Bbb Z[\sqrt 5]\,$ is then maximal and thus prime.

DonAntonio

 

[mathwonk]

i guess the first step for me would be to try to find a square root of -5 in the ring Z11.