Exercise 8.12

Proof. Suppose that $p$ is prime, $p\equiv 1(\mathit{mod}4)$, and $p=a^2+b^2=c^2+d^2$, where $a,b,c,d$ are positive integers, $a,c$ odd, $b,d$ even. We will show that $a=c,b=d$.

As $p=N(a+\mathit{bi})$, $\pi =a+\mathit{bi}$ is irreducible in $\mathbb{Z}[i]$ : indeed $\pi =\mathit{uv}$ implies that $p=N(\pi )=N(u)N(v)$, so $N(u)=1$ or $N(v)=1$, and $u$ or $v$ is an unit.

Since $\mathbb{Z}[i]$ is a principal ideal domain, $\pi$ is a prime in $\mathbb{Z}[i]$.

$(a+\mathit{bi})(a-\mathit{bi})=(c+\mathit{di})(c-\mathit{di})$, so the prime $\pi$ divides $c+\mathit{di}$, or it divides $c-\mathit{di}$.

As $N(\pi )=N(c+\mathit{di})=N(c-\mathit{di})$, the quotient is an unit. Therefore $\pi$ is an associate of $c+\mathit{di}$ or $c-\mathit{di}$. Since the units in $\mathbb{Z}[i]$ are $1,-1,i,-i$,

In all cases, $a=\pm c,b=\pm d$, or $a=\pm d,b=\pm c$. Since $a,b,c,d$ are positive, $a=c,b=d$, or $a=d,b=c$. As $a,c$ are odds, and $b,d$ even, $a=c,b=d$ : the unicity of the decomposition is proved. □