Number Theory - Sum of squares part question

Question

So if $p$ is a prime number and $p = 1$ mod $4$. So I shows that $[-1]_p$ has a square root in the field $\mathbb{Z}/p\mathbb{Z}$ and also that there is some $n$ such that p divides $n^2+1$.

Now I am having trouble taking n from the above part and showing that $\mathbb{Z}[i]$, $p$ divides neither $n + i$ nor $n- i$ Conclude that $p$ is not an irreducible element of $\mathbb{Z}[i]$.

http://math.stackexchange.com/questions/122048/1-is-a-quadratic-residue-modulo-p-if-and-only-if-p-equiv-1-pmod4 — lab bhattacharjee, Oct 17 '13 at 12:03
If $p$ were to divide $n+i$, the quotient would be $(n/p)+(1/p)i$, but that's not in ${\bf Z}[i]$, which is ${,a+bi:a,b{\rm\ in\ }{\bf Z},}$. — Gerry Myerson, Oct 17 '13 at 12:24

score 1 · Answer 1 · answered Oct 17 '13 at 12:33

1

For example, suppose

$$p\mid(n+i)\implies n+i=p(a+bi)=pa+pbi\;,\;\;a,b\in\Bbb Z\implies pb=1$$

and the last equality is impossible.

answered Oct 17 '13 at 12:33

DonAntonio

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Number Theory - Sum of squares part question

1 Answers1