## 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

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

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 [!].