I'm reading the english translation of Algebraic Number Theory by "Jurgen Neukirch" and am stuck on a proof in the introduction. It seems there is some "obvious" fact i'm missing.
Here is the situation:
On page 2 we are working on a proof that every prime $p \equiv 1 \mod 4$ can be decomposed into a sum of 2 squares.
We have shown that the ring $\mathbb{Z}[i]$ is Euclidean, and we will show that primes $p \equiv 1 \mod 4$ do not remain prime, so whats left to show is that assume a prime $p \equiv 1 \mod 4$, is not a prime in $\mathbb{Z}[i]$, can we then prove it is a sum of 2 squares.
The book outlines the following proof:
$$ p = a b $$ where $a,b$ are non unit elements of $\mathbb{Z}[i]$. Now define $$N(z) = |z|^2$$
Then it follows that $$p^2 = N(a)N(b)$$ The observation that $$N(a),N(b) \ne 1$$ is made. Then the book jumps to conclusion $$p = N(a)$$ and from the the remainder of the proof is trivial for showing $p$ is a sum of 2 squares.
I do not understand at all how $p=N(a)$ can be deduced if $N(b)$ is not equal to 1, and $p^2 = N(a)N(b)$. Attempts to analyze it get messy since I have a non-unitary $N(b)$, so $p = \sqrt{N(a)N(b)}$ and from here next steps aren't obvious.
Neukirch argues that both $N\left(a\right)$ and $N\left(b\right)$ are $\neq 1$. But $N\left(a\right)$ and $N\left(b\right)$ are two nonnegative integers (since the norm of any Gaussian integer is a nonnegative integer) whose product is $N\left(a\right) N\left(b\right) = N\left(ab\right) = p^2$. Since $p$ is prime, this can only happen in one of the following three cases:
Case 1: We have $N\left(a\right) = 1$ and $N\left(b\right) = p^2$.
Case 2: We have $N\left(a\right) = p$ and $N\left(b\right) = p$.
Case 3: We have $N\left(a\right) = p^2$ and $N\left(b\right) = 1$.
Since Cases 1 and 3 cannot hold (because $N\left(a\right)$ and $N\left(b\right)$ are $\neq 1$), we thus conclude that Case 2 must hold. In other words, $N\left(a\right) = p$ and $N\left(b\right) = p$. Thus, $p = N\left(a\right) = \left(\operatorname{Re} a\right)^2 + \left(\operatorname{Im} a\right)^2$. This is a representation of $p$ as a sum of two squares of integers.