# Properties

 Label 23.13.0.1 Base $$\Q_{23}$$ Degree $$13$$ e $$1$$ f $$13$$ c $$0$$ Galois group $C_{13}$ (as 13T1)

# Related objects

## Defining polynomial

 $$x^{13} - 5 x + 3$$

## Invariants

 Base field: $\Q_{23}$ Degree $d$ : $13$ Ramification exponent $e$ : $1$ Residue field degree $f$ : $13$ Discriminant exponent $c$ : $0$ Discriminant root field: $\Q_{23}$ Root number: $1$ $|\Gal(K/\Q_{ 23 })|$: $13$ This field is Galois and abelian over $\Q_{23}$.

## Intermediate fields

 The extension is primitive: there are no intermediate fields between this field and $\Q_{ 23 }$.

## Unramified/totally ramified tower

 Unramified subfield: 23.13.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of $$x^{13} - 5 x + 3$$ Relative Eisenstein polynomial: $x - 23 \in\Q_{23}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_{13}$ (as 13T1) Inertia group: Trivial Unramified degree: $13$ Tame degree: $1$ Wild slopes: None Galois mean slope: $0$ Galois splitting model: $x^{13} - x^{12} - 60 x^{11} + 27 x^{10} + 1199 x^{9} - 33 x^{8} - 9610 x^{7} - 3352 x^{6} + 33548 x^{5} + 20328 x^{4} - 47723 x^{3} - 34869 x^{2} + 21271 x + 15667$