Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of extensions of local fields

Introduction

In [Sinclair: Algorithms for Enumerating Invariants and Extensions of Local Fields] we give formulas for the number of extensions of a $$(\pi)$$-adic field with given, inertia degree, ramification index, discriminant, ramification polygon, and residual polynomials of the segments of the ramification polygon. The tables presented here illustrate how the extension of a given degree are distributed over the classes with different invariants.

In the tables the column Extensions gives the number of extensions with the given invariants. The last of these numbers is obtained by Krasner's mass formula [Nombre des extensions d’un degré donné d’un corps p-adique, 1966].

Invariants

We recall the definitions of the invariants of the extensions given in the tables.

Discriminant $$(\pi)^{n+j-1}$$

The possible discriminants of extensions of a given degree are described by Ore's conditions [Bemerkungen zur Theorie der Differente, 1926]:

Proposition. Let $$K$$ be a finite extension of $$\mathbb{Q}_p$$, $$\mathcal{O}_K$$ its valuation ring with maximal ideal $$(\pi)$$. Denote by $$v_\pi$$ the valuation on $$K$$ that is normalized such that $$v_\pi(\pi)=1$$. Given $$j \in \mathbb{N}$$ let $$a,b\in\mathbb{N}$$ be such that $$j = an + b$$ with $$0 \leq b < n$$. Then there exist totally ramified extensions $$L/K$$ of degree $$n$$ and discriminant $$(\pi)^{n+j-1}$$ if and only if $$ \min\{v_\pi(b) n, v_\pi(n) n\} \leq j \leq v_\pi(n) n. $$

In the tables in the column j discriminants are repesented by $$j$$.

Ramification Polygons

Definition. Let $$K$$ be a finite extension of $$\mathbb{Q}_p$$, $$\mathcal{O}_K$$ its valuation ring. Let $$\varphi\in\mathcal{O}_K[x]$$ be Eisenstein and denote by $$\alpha$$ a root of $$\varphi$$. The ramification polygon of $$\varphi$$ is the Newton polygon of the ramification polynomial $$\rho(x)=\varphi(\alpha x + \alpha)/(\alpha^n)\in \mathbb{Q}_p(\alpha)[x]$$ of $$\varphi$$.

Proposition. The ramification polygon P of $$\varphi$$ is an invariant of $$L=K(\alpha)$$ called the ramification polygon of $$L$$.

The slopes of the segments of P are the (generalized) lower ramification breaks of $$L/K$$.

In the column Ramification Polygon the ramification polygon as a set of vertices. Slopes contains the slopes of the segments of the ramification polygon.

Residual Polynomials

Residual (or associated) polynomials were introduced by Ore [Newtonsche Polygone in der Theorie der algebraischen Körper (1928)].

Definition. Let $$L$$ be a finite extension of $$K$$ with uniformizer $$\alpha$$ and denote by $$v_\alpha$$ the (exponential) valuation that is normalized such that $$v_\alpha(\alpha)=1$$. Let $$\rho(x)=\sum_i \rho_i x^i\in\mathcal{O}_L[x]$$. Let S be a segment of the Newton polygon of $$\rho$$ of length $$l$$ with end points $$(k,v_\alpha(\rho))$$ and $$(k+l,v_\alpha(\rho_{k+l}))$$, and slope $$-h/e=\left(v_\alpha(\rho_{k+l})-v_\alpha(\rho_k)\right)/l$$ then

$$ A(x)=\sum_{j=0}^{l/e}\underline{\rho_{je+k}\alpha^{jh-v_\alpha(\rho_k)}}x^{j}\in\underline{L}[x] $$

where $$ \underline{L}$$ denotes the residue class field of $$L$$ is called the residual polynomial of S.

Although the set of residual polynomials of the segments of the ramification polygon $$\varphi$$ is not an invariant of $$L/K$$ it can be used to define an invariant.

Theorem. Let $$S_1,\dots,S_\ell$$ be the segments of the ramification polygon of an Eisenstein polynomial $$\varphi\in\mathcal{O}_K[x]$$. For $$1\le i\le \ell$$ let $$-h_i/e_i$$ be the slope of $$S_i$$ and $$A_i(x)$$ its residual polynomial. Then

$$ \mathcal{A}= \left\{ \left(\gamma_{\delta,1}{A_1}(\underline\delta^{h_1} x),\dots, \gamma_{\delta,\ell}{A_\ell}(\underline\delta^{h_\ell} x)\right) : \underline\delta\in\underline{K}^\times \right\} $$

where $$ \gamma_{\delta,\ell}=\delta^{-h_\ell\deg A_\ell}, $$ and $$ \gamma_{\delta,i}=\gamma_{\delta,i+1}\delta^{-h_i\deg A_i} $$ for $$1\le i\le \ell-1$$ is an invariant of the extension $$K[x]/(\varphi)$$. We call $$ \mathcal{A}$$ the residual polynomial classes of the ramification polygon of $$\varphi$$.

In the column Residual Polynomials we give one representative of the invariant $$\mathcal{A}$$. The column #A contains the number of tuples of polynomials in $$\mathcal{A}$$.

Polynomials

The algorithm in [Pauli and Sinclair: Enumerating Extensions of $$(\pi)$$-Adic Fields with Given Invariants] in many cases constructs a unique generating polynomial for each extension with the given invariants.

We give the number of polynomials constructed by the algorithm in the column Polynomials. When to obtain unique generating polynomials, polynomials that generate the same extension as another polynomial have to be filtered out, this is indicated by [!].