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

I noticed that [itex]\mathbb{Z}[\sqrt{-5}][/itex] 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

[itex]\mathbb{Z}[\sqrt{-5}]/(11)\cong\mathbb{Z}_{11}[x]/(x^2+1)[/itex]

If this is true then I can conclude that [itex]\mathbb{Z}[\sqrt{-5}][/itex] is a domain because

[itex]\mathbb{Z}_{11}[x]/(x^2+1)[/itex] is a finite field.

Thank you

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

I noticed that [itex]\mathbb{Z}[\sqrt{-5}][/itex] 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

[itex]\mathbb{Z}[\sqrt{-5}]/(11)\cong\mathbb{Z}_{11}[x]/(x^2+1)[/itex]

If this is true then I can conclude that [itex]\mathbb{Z}[\sqrt{-5}][/itex] is a domain because

[itex]\mathbb{Z}_{11}[x]/(x^2+1)[/itex] is a finite field.

Thank you

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

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