shy gull
?r
#import "@preview/ctheorems:1.1.3": *
#let proof = thmproof("proof", "Proof")
#proof[
Define a homomorphism $phi: ZZ[x] -> ZZ[i], phi(f(x)) = f(i) => im(phi) = ZZ[i]$.
By the Fundamental Homomorphism Theorem, we have $ZZ[i] =e ZZ[x] slash ker(phi)$.
As $phi(x^2 + 1) = i^2 + 1 = 0$ so $x^2 + 1 in ker(phi)$.
But $(forall z in ZZ[x])phi(z) != 0$ if $deg(z) = 0 or 1$.
So $x^2 + 1$ is a non-zero polynomial in $ker(phi)$ of minimal degree.
Therefore, $ker(phi) = (x^2 + 1)$.
// $therefore ZZ[x] slash (x^2 + 1) =e ZZ[i]$.
]
old badger
hazy shoal
