Let $R=\mathbb{Z}[i]/(3+i)$. Why is this a ring and show that $|R|=10$.

Question

Let $R=\mathbb{Z}[i]/(3+i)$. Why is this a ring and show that $|R|=10$. I'm not allowed to use isomorphisms for the second bit.

For the first bit I said $(3+i)$ is an ideal, $\mathbb{Z}[i]$ is a ring, and $\mathbb{Z}[i]/(3+i)$ is a quotient group so it is a ring.

Second bit no idea.

Think of this as polynomials with integer coefficients which then get evaluated at the complex number i. Why is i^3 equal to -i equal to 3 in this ring? — Tac-Tics, Mar 06 '16 at 00:47
$R=(3+i)+r $where $r$ is in $\mathbb{Z}[i]$. If $3=-i$ then $a+bi=a-b(3)=a-3b$? — Polp, Mar 06 '16 at 01:06
When you quotient out by something you are effectively treating that "thing" as "zero". In other words, $\mathbb{Z}[i]/(3+i)$ is the structure you get by taking $\mathbb{Z}[i]$ and imposing $3+i=0$. — ocg, Mar 06 '16 at 01:15
I think $(3+i)$ is an ideal, does it still mean only $3+i=0$? — Polp, Mar 06 '16 at 01:16
$10=0$ and $i=7$, by squaring. Can you see that this leaves us with only $10$ distinct elements? — ocg, Mar 06 '16 at 01:28
Nope it is in my lecture notes. Just states that it is a ring and $|R|=10$ so I was wondering why. — Polp, Mar 06 '16 at 01:41
The only distinct elements are $0$ to $9$. But what about the imaginary parts — Polp, Mar 06 '16 at 01:43

score 3 · Accepted Answer · answered Mar 06 '16 at 01:46

I like to think of a quotient thing as we keep everything but modify the equality on the given structure.
So, things like (some ring)$/(a-b)$ translate to [$a=b$ in the new structure].

Now, the original structure is $\Bbb Z[i]$, the Gaussian integers.

Then we modify the equality so that it respects the structure ($+,-,\cdot$), by posing $3=-i$.

In the new structure therefore every Gaussian integer $a+bi$ will be equal to the integer $a-3b$.

We get the integers $0,1,2,\dots,9$, and, as $3^2=(-i)^2$, we have $10=0$.

Now the only thing left to prove is that these ten numbers are still different in the quotient ring.

Btw, observe that $(3-i)\overline{(3-i)}=3^2+1^2=10$. – Berci Mar 06 '16 at 01:51 — Berci, Mar 06 '16 at 01:51

Let $R=\mathbb{Z}[i]/(3+i)$. Why is this a ring and show that $|R|=10$.

1 Answers1