Neukirch

Here are my exercise solutions and notes for Neukirch’s Algebraic Number Theory.

1.1. Recall that $(ab)(\overline{ab})=(a\overline{a})(b\overline{b})$ for $a,b\in\mathbb{C}$ . Therefore $N(ab)=N(a)N(b)$ for $a,b\in\mathbb{Z}[i]$ . Moreover, $N(a)\in\mathbb{Z}_{\ge 0}$ for all $a\in\mathbb{Z}[i]$ . Thus if $ab=1$ we must have $N(a)N(b)=1$ and therefore $N(a)=N(b)=1$ .

1.2. Suppose that $\alpha\beta=\epsilon\gamma^n$ for $\alpha,\beta,\gamma,\epsilon\in\mathbb{Z}[i]$ with $\epsilon$ a unit and $\alpha,\beta$ relatively prime. Since $\mathbb{Z}[i]$ is a UFD we may write out the prime factorizations $\alpha=\epsilon&39;p_1^{a_1}\ldots p_r^{a_r}$ and $\beta=\epsilon&39;&39;q_1^{b_1} \ldots q_{\ell}^{b_\ell}$ with $\epsilon&39;,\epsilon&39;&39;$ units and the $p_i$ ’s and $q_j$ ’s prime. Since $\alpha,\beta$ are relatively prime, the $p_i$ ’s and $q_j$ ’s are all distinct primes. Therefore, since $\alpha\beta=\epsilon\gamma^n$ , we must have $n\mid a_i$ and $n\mid b_j$ for all $i,j$ . I.e., we have $\alpha=\epsilon&39;\xi^n$ and $\beta=\epsilon&39;&39;\eta$ where $\xi=p_1^{a_1/n}\ldots p_r^{a_r/n}$ and $\eta=q_1^{b_1/n}\ldots q_\ell^{b_\ell/n}$ .

1.3. Observe that if $x^2+y^2=z^2$ for $x,y,z\in\mathbb{Z}$ , then $\alpha\overline{\alpha}=z^2$ for $\alpha=x+iy$ . Therefore by Exercise 1.2 we have $x+iy=\epsilon\xi^2$ with $\epsilon$ a unit and $\xi=u+iv$ . Therefore $x+iy=\epsilon(u^2-v^2+2uvi)$ . So $x=\epsilon(u^2-v^2)$ and $y=\epsilon 2uv$ . From Exercise 1 we see that the only units in $\mathbb{Z}[i]$ are $\pm 1,\pm i$ . We cannot have $\epsilon=\pm i$ as $x,y\in\mathbb{Z}$ . Thus we may ignore the $\epsilon$ as it just swaps the sign.

Observe that if $(u,v)>1$ then we we would have $(x,y)>1$ , a contradiction. If $u,v$ are both odd, then $u^2-v^2\equiv 0(\text{mod } 3)$ and $2uv\equiv 2(\text{mod } 3)$ . Thus $z^2\equiv 2 (\text{mod } 3)$ , a contradiction.

1.4. Recall that an ordering on a ring satisfies $a\le b$ implies $a+c\le b+c$ for all $a,b,c\in R$ , as well as $0\le a$ and $0\le b$ implies $0\le ab$ for all $a,b\in R$ . The first implication shows that $-a\le 0\le a$ for all $a\ge 0$ ; in particular $-1\le 0\le 1$ . If $i\ge 0$ , then $i^2=-1\le 0$ , a contradiction. However, if $i\le 0$ then $-i\ge 0$ and $(-i)^2=-1\le 0$ .

1.5. Observe that we may define a norm on $\mathbb{Z}[\sqrt{-d}]$ by $N(\alpha)=\alpha\overline{\alpha}$ where if $\alpha=a+b\sqrt{-d}$ we have $\overline{\alpha}=a-b\sqrt{-d}$ . This gives $N(a+b\sqrt{-d})=a^2+b^2d$ . Again we see that $N(\alpha\beta)=N(\alpha)N(\beta)$ and thus an element $\alpha$ is a unit if and only if $N(\alpha)=1$ . The only elements of $\mathbb{Z}[\sqrt{-d}]$ with norm $1$ are $\pm 1$ .

1.6. After struggling for a while and digging a bit on math stack exchange, it appears that this question is fairly difficult for its place in the book. Perhaps I will revisit this question after progressing further.