# Properties

 Label 47.14.0.1 Base $$\Q_{47}$$ Degree $$14$$ e $$1$$ f $$14$$ c $$0$$ Galois group $C_{14}$ (as 14T1)

# Related objects

## Defining polynomial

 $$x^{14} - x + 23$$

## Invariants

 Base field: $\Q_{47}$ Degree $d$: $14$ Ramification exponent $e$: $1$ Residue field degree $f$: $14$ Discriminant exponent $c$: $0$ Discriminant root field: $\Q_{47}(\sqrt{5})$ Root number: $1$ $|\Gal(K/\Q_{ 47 })|$: $14$ This field is Galois and abelian over $\Q_{47}.$

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Unramified/totally ramified tower

 Unramified subfield: 47.14.0.1 $\cong \Q_{47}(t)$ where $t$ is a root of $$x^{14} - x + 23$$ Relative Eisenstein polynomial: $$x - 47$$$\ \in\Q_{47}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_{14}$ (as 14T1) Inertia group: trivial Unramified degree: $14$ Tame degree: $1$ Wild slopes: None Galois mean slope: $0$ Galois splitting model: $x^{14} - x^{13} - 13 x^{12} + 12 x^{11} + 66 x^{10} - 55 x^{9} - 165 x^{8} + 120 x^{7} + 210 x^{6} - 126 x^{5} - 126 x^{4} + 56 x^{3} + 28 x^{2} - 7 x - 1$