# Sub-quotients and Gaussian Integer rings

• Feryll
In summary, the author is trying to determine when a prime ideal generated by a prime p is prime in Z[i], and is using Z[i]/(p) as a way to determine when this is the case.
Feryll
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 is exactly.

p is a prime.

"We know that p is reducible [ie p=(a+bi)(a-bi), a,b∈Z] iff (p)=pZ is not prime.
Consider the isomorphisms
Z≅Z[X]/(X2+1)
Z/(p)≅Z[X]/(X2+1,p)
Z/(p)≅(Z[X]/(p))/(X2+1)
Z/(p)≅Fp[X]/(X2+1)

If p≠2, we have
(p) reducible⇔X2+1 factors in Fp[X]
⇔-1∈(Fp*)2, the group of squares in Fp*
⇔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 Fp[X]?

(p) is the ideal generated by p in Z.

Z[X] and F_p[X] are the rings of polynomials in the indeterminate X with coefficients in the rings Z and F_p, resp.

What your book is doing is showing that just because p is a prime in Z (so the ideal generated by p in Z is a prime ideal), it doesn't follow that the ideal generated by p is prime in Z. The book is (essentially) trying to determine when this is the case, which is precisely when the quotient Z/(p) is a domain.

P.S. You probably don't mean to say "subquotient" here; "quotient" will do just fine.

morphism said:
(p) is the ideal generated by p in Z.

Z[X] and F_p[X] are the rings of polynomials in the indeterminate X with coefficients in the rings Z and F_p, resp.

What your book is doing is showing that just because p is a prime in Z (so the ideal generated by p in Z is a prime ideal), it doesn't follow that the ideal generated by p is prime in Z. The book is (essentially) trying to determine when this is the case, which is precisely when the quotient Z/(p) is a domain.

P.S. You probably don't mean to say "subquotient" here; "quotient" will do just fine.

Thanks, you really cleared up some concepts, then. Also, yes, I must have meant quotient. Not sure where the sub came from.

## 1. What are sub-quotients in relation to Gaussian Integer rings?

Sub-quotients are elements of a quotient group obtained by dividing a larger group by a subgroup. In the context of Gaussian Integer rings, sub-quotients are obtained by dividing the ring of Gaussian Integers by a non-zero ideal.

## 2. How are sub-quotients calculated in Gaussian Integer rings?

In order to calculate sub-quotients in Gaussian Integer rings, one must first identify a non-zero ideal. Then, the ideal must be divided into the ring of Gaussian Integers. The resulting elements are the sub-quotients.

## 3. What is the significance of sub-quotients in Gaussian Integer rings?

Sub-quotients are important in the study of Gaussian Integer rings because they provide a way to understand the structure and properties of the ring. They also play a key role in the classification of ideals and in determining the factorization of Gaussian Integers.

## 4. Can sub-quotients of Gaussian Integer rings be used in practical applications?

Yes, sub-quotients of Gaussian Integer rings have practical applications in areas such as coding theory, cryptography, and number theory. They can be used to construct error-correcting codes, generate random numbers, and in the encryption and decryption of data.

## 5. How do sub-quotients in Gaussian Integer rings relate to complex numbers?

Gaussian Integers are a subset of complex numbers, so sub-quotients in Gaussian Integer rings can be thought of as sub-quotients of the complex numbers. However, sub-quotients in Gaussian Integer rings have additional properties and structure that make them distinct from sub-quotients of the complex numbers.

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