This has to do with number theory along with group and set theory, but the main focus of the proof is number theory, so forgive me if I'm in the wrong place. I've been struggling to understand a piece of a proof put forth in my book. I know what the Gaussian integers are exactly, and what a subquotient group and isomorphism is (although probably not perfectly, right?), but I don't know what pZ(adsbygoogle = window.adsbygoogle || []).push({}); is exactly.

p is a prime.

"We know that p is reducible [ie p=(a+bi)(a-bi), a,b∈Z] iff (p)=pZis not prime.

Consider the isomorphisms

Z≅Z[X]/(X^{2}+1)

Z/(p)≅Z[X]/(X^{2}+1,p)

Z/(p)≅(Z[X]/(p))/(X^{2}+1)

Z/(p)≅F_{p}[X]/(X^{2}+1)

If p≠2, we have

(p) reducible⇔X"^{2}+1 factors in F_{p}[X]

⇔-1∈(F_{p}^{*})^{2}, the group of squares in F_{p}^{*}

⇔p congruent to 1 mod(4) (Euler's criterion)

I just barely know where to start with what he's getting at. What is (p), exactly? What is Z[X] and F_{p}[X]?

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# Sub-quotients and Gaussian Integer rings

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