# Properties

 Label 5.10.5.1 Base $$\Q_{5}$$ Degree $$10$$ e $$2$$ f $$5$$ c $$5$$ Galois group $C_{10}$ (as 10T1)

# Learn more about

## Defining polynomial

 $$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$ ## Invariants

 Base field: $\Q_{5}$ Degree $d$: $10$ Ramification exponent $e$: $2$ Residue field degree $f$: $5$ Discriminant exponent $c$: $5$ Discriminant root field: $\Q_{5}(\sqrt{5})$ Root number: $1$ $|\Gal(K/\Q_{ 5 })|$: $10$ This field is Galois and abelian over $\Q_{5}.$

## 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: 5.5.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of $$x^{5} - x + 2$$ Relative Eisenstein polynomial: $$x^{2} - 5$$$\ \in\Q_{5}(t)[x]$ ## Invariants of the Galois closure

 Galois group: $C_{10}$ (as 10T1) Inertia group: Intransitive group isomorphic to $C_2$ Unramified degree: $5$ Tame degree: $2$ Wild slopes: None Galois mean slope: $1/2$ Galois splitting model: $x^{10} - x^{9} - 13 x^{8} + 8 x^{7} + 46 x^{6} - 11 x^{5} - 52 x^{4} + 7 x^{3} + 18 x^{2} - 3 x - 1$