# $p$-adic fields downloaded from the LMFDB on 15 July 2024.
# Search link: https://www.lmfdb.org/padicField/?p=19&n=4
# Query "{'p': 19, 'n': 4}" returned 5 fields, sorted by prime.
# Each entry in the following data list has the form:
# [Label, Polynomial, $p$, $e$, $f$, $c$, Galois group, Slope content]
# For more details, see the definitions at the bottom of the file.
"19.4.0.1" [2, 11, 2, 0, 1] 19 1 4 0 [4, 1] [[], 1, 4]
"19.4.2.1" [6445, 756, 366, 36, 1] 19 2 2 2 [4, 2] [[], 2, 2]
"19.4.2.2" [722, -76532, -767106, -2888, 1] 19 2 2 2 [4, 1] [[], 2, 2]
"19.4.3.1" [19, 0, 0, 0, 1] 19 4 1 3 [4, 3] [[], 4, 2]
"19.4.3.2" [38, 0, 0, 0, 1] 19 4 1 3 [4, 3] [[], 4, 2]
# Label --
# Each $p$-adic field $K$ has a unique label of the form "p.n.c.num", where
# - p is the residue field characteristic,
# - n is the degree of $K$ over $\mathbb{Q}_p$,
# - c is the discriminant exponent of $K$,
# - num is a positive integer giving the index of $K$ in a standard ordered list of those with the same triple (p,n,c).
#Polynomial (coeffs) --
# The **defining polynomial** of a $p$-adic field $K$ is an irreducible polynomial $f(x)\in\mathbb{Q}_p[x]$ such that $K\cong \mathbb{Q}_p(a)$, where $a$ is a root of $f(x)$.
# The defining polynomial can be chosen to be monic with coefficients in $\mathbb{Z}_p$; by Krasner's lemma, we can further take $f(x)\in \mathbb{Z}[x]$.
#$p$ (p) --
# If $p$ is a prime number, then every rational number $r$ can be written in the form $r=p^j \frac{a}{b}$ where $\gcd(p, ab)=1$. We define $|r|_p = p^{-j}$, and define a metric on $\Q$ by $d(u,v) = |u-v|_p$; in other words, $u$ and $v$ are close to each other if they are congruent modulo a larger power of $p$.
# The completion of $\Q$ with respect to this metric is $\Q_p$, which is itself a field in the natural way. Any non-zero element $x \in \Q_p$ can be written uniquely as the sum of a convergent series $x = \sum_{n=n_0}^{+\infty} a_n p^n$, where $a_n \in \{ 0, \cdots, p-1 \}$, $n_0 \in \Z$ is such that $|x|_p=p^{-n_0}$, and $a_{n_0} \neq 0$.
# Example: the element $a = 3 + 7 + 2 \cdot 7^2 + 6 \cdot 7^3 + 7^4 + 2 \cdot 7^5 + \cdots$ satisfies $a^2=2$, since truncating the expansion after $n$ terms and squaring yields a number which is congruent to $2 \bmod 7^n$.
#$e$ (e) --
# If $F$ is a finite extension of $\Q_p$ and $K$ a finite extension of $F$. Then $\mathcal{O}_F$ and $\mathcal{O}_K$, the ring of integers of $F$ and $K$ are discrete valuation domains, so they have unique maximal ideals $P_F$ and $P_K$ which are principal. If $P_F=(\pi_F)$, the element $\pi_F$ is a **uniformizer** for $F$.
# The principal ideal $\pi_F\mathcal{O}_K=P_K^e$ for some positive integer $e$. The integer $e$ is the **ramification index** for $K$ over $F$. The ramification index of $K$ is then the ramification index for $K$ over $\Q_p$.
# If $e=1$, then we say that the extension is **unramified**, and if $e=[K:\Q_p]$, then we say that the extension is **totally ramified**.
#$f$ (f) --
# The **residue field degree** of a nonarchimedean local field is the degree of its residue field as an extension of its prime field.
#$c$ (c) --
# The **discriminant** of a $p$-adic field $K$ is the square of the determinant of the matrix
# \[
# \left( \begin{array}{ccc}
# \sigma_1(\beta_1) & \cdots & \sigma_1(\beta_n) \\
# \vdots & & \vdots \\
# \sigma_n(\beta_1) & \cdots & \sigma_n(\beta_n) \\
# \end{array} \right)
# \]
# where $\sigma_1,..., \sigma_n$ are the embeddings of $K$ into an algebraic closure $\overline{\mathbb{Q}}_p$, and $\{\beta_1, \ldots, \beta_n\}$ is an integral basis for the ring of integers of $K$.
# The discriminant of $K$ is an element of $\mathbb{Z}_p$ which is well-defined up to the square of a unit. Thus, it is of the form $p^c u$ where $u$ is a unit. The value $c$ is the discriminant exponent for $K$. Together with the discriminant root field of $K$, it determines the discriminant of $K$ (up to the square of a unit).
#Galois group (gal) --
# Let $K$ be a finite degree $n$ separable extension of a field $F$, and $K^{gal}$ be its
# Galois (or normal) closure.
# The **Galois group** for $K/F$ is the automorphism group $\Aut(K^{gal}/F)$.
# This automorphism group acts on the $n$ embeddings $K\hookrightarrow K^{gal}$ via composition. As a result, we get an injection $\Aut(K^{gal}/F)\hookrightarrow S_n$, which is well-defined up to the labelling of the $n$ embeddings, which corresponds to being well-defined up to conjugation in $S_n$.
# We use the notation $\Gal(K/F)$ for $\Aut(K/F)$ when $K=K^{gal}$.
# There is a naming convention for Galois groups up to degree $47$.
#Slope content (slopes) --
# Let $G$ be the Galois group of the Galois closure $K^{gal}$ of a $p$-adic field $K$. Then $G$ has a filtration by higher ramification groups, which in turn are connected to the discriminants of subfields of $K^{gal}$. The *slope content* encodes information about this filtration giving the location and sizes of the jumps.
# Let $G^i$ be the $i$th higher ramification group in upper numbering from Serre's *Local Fields* \cite{doi:10.1007/978-1-4757-5673-9}. They form a descending filtration of normal subgroups where $i$ is a continuous parameter, $i\geq -1$. Let $G^{(j)} = G^{j-1}$, and $G^{(j+)} = \cup_{\epsilon>0} G^{(j+\epsilon)}$. Then
# - $G^{(0)}=G$.
# - $G^{(1)}$ is the inertia subgroup of $G$, i.e., the kernel of the natural map from $G$ to the Galois group of the residue field.
# - the fixed field of $G^{(1)}$ is the maximum unramified subfield of $K^{gal}/\mathbb{Q}_p$, which we denote by $K^{unram}$. The degree $[K^{unram}:\mathbb{Q}_p]$ is the unramified degree for $K^{gal}/\mathbb{Q}_p$. The extension $K^{gal}/K^{unram}$ is totally ramified.
# - the group $G^{(1)}/G^{(1+)}$ is cyclic of order prime to $p$. Its order is the tame degree for $K^{gal}/\mathbb{Q}_p$ as it is the degree of the largest tamely totally ramified subextension.
# - The group $G^{(1+)}$ is a $p$-group. Values of $s>1$ giving jumps in the filtration $G^{(s)}\neq G^{(s+)}$ are wild slopes. We repeat a slope $m$ times if $p^m =[G^{(s)} : G^{(s+)}]$, and then list the wild slopes in non-decreasing order.
# The **slope content** then takes the form $[s_1,\ldots, s_k]_t^u$ where $t$ is the tame degree, $u$ is the degree of the maximum unramified subfield, and the $s_i$ give the list of the wild slopes. In particular, $[K^{gal}:\Q_p]=tup^k$. If $t$ or $u$ is equal to $1$, it is not printed.