# Thread: Quotient Ring of the Gaussian Integers

1. ## Quotient Ring of the Gaussian Integers

Given $\displaystyle p \equiv 3 \mod{4}$ is prime in $\displaystyle \mathbb{Z}$, show $\displaystyle \mathbb{Z}[i]/(p)$ has order $\displaystyle p^2$.

2. Originally Posted by chiph588@
Given $\displaystyle p \equiv 3 \mod{4}$ is prime in $\displaystyle \mathbb{Z}$, show $\displaystyle \mathbb{Z}[i]/(p)$ has order $\displaystyle p^2$.
An odd prime p has a form either 1(mod 4) or 3(mod 4). We can see that 1(mod 4) is reducible in Z[i] (For example, 5=(1+2i)(1-2i)). An odd prime p is irreducible iff it has the form 3(mod 4).

We know that Z[i] is an integral domain and a PID. We also know that $\displaystyle p \equiv 3 (mod{4})$ is irreducible. Thus <p> is maximal ideal in Z[i]. It follows that Z[i]/<p> is a field having p^2 elements.

3. Originally Posted by aliceinwonderland
It follows that Z[i]/<p> is a field having p^2 elements.
I don't quite see how that follows.

4. Originally Posted by chiph588@
I don't quite see how that follows.
An element of Z[i]/<p> has the form a+bi. The number of choices of a is p and the number of choices for b is p. So it has p^2 elements.

For example, let p =3. Then, Z[i]/<3>= {0, 1,2, i, 2i, 1+i, 1+2i, 2+i, 2+2i}. It has 9 elements. You can check other cases and verify them.

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### find no of elements in a quotient ring of gauszian intevers

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