# Properties

 Label 23.14.0.1 Base $$\Q_{23}$$ Degree $$14$$ e $$1$$ f $$14$$ c $$0$$ Galois group $C_{14}$ (as 14T1)

# Related objects

## Defining polynomial

 $$x^{14} - x + 7$$ ## Invariants

 Base field: $\Q_{23}$ Degree $d$: $14$ Ramification exponent $e$: $1$ Residue field degree $f$: $14$ Discriminant exponent $c$: $0$ Discriminant root field: $\Q_{23}(\sqrt{5})$ Root number: $1$ $|\Gal(K/\Q_{ 23 })|$: $14$ This field is Galois and abelian over $\Q_{23}.$

## 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: 23.14.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of $$x^{14} - x + 7$$ Relative Eisenstein polynomial: $$x - 23$$$\ \in\Q_{23}(t)[x]$ ## Invariants of the Galois closure

 Galois group: $C_{14}$ (as 14T1) Inertia group: trivial Wild inertia group: $C_1$ Unramified degree: $14$ Tame degree: $1$ Wild slopes: None Galois mean slope: $0$ Galois splitting model: $x^{14} - x^{13} + 3 x^{12} - 11 x^{11} + 44 x^{10} + 156 x^{9} - 250 x^{8} - 749 x^{7} + 1560 x^{6} + 4490 x^{5} + 3202 x^{4} + 755 x^{3} + 4050 x^{2} + 1750 x + 625$