What is the purpose of this proposition?

Question

At the end of chapter 5 of stein's book A Computational Introduction to Algebraic Number Theory he proves proposition 5.2.4 which states that:

Given a prime ideal $\mathfrak{p}$ in a Dedekind domain $R$ we have the isomorphism $$ \frac{\mathfrak{p}^n}{\mathfrak{p}^{n+1}} \cong \frac{R}{\mathfrak{p}} $$ of $R$-modules for any $n \geq 0$.

What is the point of including this proposition, other than the fact it can be proved using the Chinese Remainder Theorem? I don't see why this is important.

Answer 1

It's used, for example, to show that the ideal norm is multiplicative. This is Prop 6.3.4 in Stein's notes.

For a number field $K$, for each non-zero ideal $\mathfrak a\subset\mathcal O_K$, we define it's norm to be $$N(\mathfrak a)=\#(\mathcal O_K/\mathfrak a).$$ It's clear from the Chinese remainder theorem that $$N(\mathfrak {ab}) = N(\mathfrak a)N(\mathfrak b)$$ if $\mathfrak {a,b}$ are coprime, so to show $N$ is multiplicative, all that remains is to show that $$N(\mathfrak p^n) = N(\mathfrak p)^n$$ for any prime ideal $\mathfrak p$. The map $$\mathcal O_K/\mathfrak p^{n+1}\to\mathcal O_K/\mathfrak p^{n}$$ has kernel $\mathfrak p^n/\mathfrak p^{n+1}$, which is isomorphic to $\mathcal O_K/\mathfrak p$ by the proposition. Hence, $$N(\mathfrak p)N(\mathfrak p^n) = N(\mathfrak p^{n+1}),$$ and the result follows by induction.

Answer 2

I don't know much about effective calculation in ANT, but it seems to me that your question is rather a general one about the "purpose" of the natural isomorphisms ¤ $\mathfrak p^n /\mathfrak p^{n+1} \cong R/\mathfrak p := k_\mathfrak p$ (the residue field at $\mathfrak p$, viewed as an additive group). Moreover it appears to be of a "local" nature. More precisely, let $R$ be the ring of integers of a number field $K$, $K_\mathfrak p$ = the completion of $K$ w.r.t. the $\mathfrak p$-adic valuation, with uniformizer $\pi$, and $R_\mathfrak p$ the ring of integers of $K_\mathfrak p$. Introduce the group of $\mathfrak p$-adic units $U_0$ and the decreasing sequence of subgroups of principal units $U_n=1 + (\pi^n)$. In addition to the isomorphisms ¤ recalled above, one has isomorphisms $U_0/U_1 \cong k_\mathfrak p ^{*}$ (obvious) and $U_n/U_{n+1} \cong \mathfrak p^n /\mathfrak p^{n+1}\cong k_\mathfrak p$ (coming obviously from ¤, or the other way around).

The principal direct (this means : without CFT) applications of these isomorphisms, I think, concern ramification , see e.g. Serre's "Local Fields", chapters IV and V. From now on, for convenience, we drop the subscript $\mathfrak p$ in the notations for $\mathfrak p$-adic local fields. For a finite Galois extension $L/K$ of such fields, with Galois group $G$, the decreasing sequence of ramification subgroups $G_i$ is defined as follows : for any integer $i\ge 0$, $G_0$ is the inertia subgroup, and $s\in G_0$ belongs to $G_i$ iff $s(\pi_L)/\pi_L\in U_i (L)$. The goal is to "unscrew" the Galois group $G$ by means of the successive quotients $G_i/G_{i+1}$. Using the aforementioned isomorphisms, one can show that the map $s\in G_i \to s(\pi_L)/\pi_L$ induces an isomorphism of $G_i/G_{i+1}$ into $U_i (L)/U_{i+1} (L)$, whose image, for certain adequate values of $i$, can be determined as the kernel of a map induced by the norm of $L/K_{nr}$, where $K_{nr}$ denotes the maximal unramified subextension of $L/K$. For the numerous information thus obtained on the filtration of the $G_i$'s, see op. cit.