# Properties

 Label 37.14.0.1 Base $$\Q_{37}$$ Degree $$14$$ e $$1$$ f $$14$$ c $$0$$ Galois group $C_{14}$ (as 14T1)

# Related objects

## Defining polynomial

 $$x^{14} - x + 22$$

## Invariants

 Base field: $\Q_{37}$ Degree $d$: $14$ Ramification exponent $e$: $1$ Residue field degree $f$: $14$ Discriminant exponent $c$: $0$ Discriminant root field: $\Q_{37}(\sqrt{*})$ Root number: $1$ $|\Gal(K/\Q_{ 37 })|$: $14$ This field is Galois and abelian over $\Q_{37}.$

## 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: 37.14.0.1 $\cong \Q_{37}(t)$ where $t$ is a root of $$x^{14} - x + 22$$ Relative Eisenstein polynomial: $x - 37 \in\Q_{37}(t)[x]$

## 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: $x^{14} - x^{13} - 13 x^{12} + 12 x^{11} + 66 x^{10} - 55 x^{9} - 165 x^{8} + 120 x^{7} + 210 x^{6} - 126 x^{5} - 126 x^{4} + 56 x^{3} + 28 x^{2} - 7 x - 1$