# Properties

 Label 5.10.10.1 Base $$\Q_{5}$$ Degree $$10$$ e $$5$$ f $$2$$ c $$10$$ Galois group $C_5^2 : C_8$ (as 10T18)

# Related objects

## Defining polynomial

 $$x^{10} + 10 x^{5} + 75 x^{2} + 25$$

## Invariants

 Base field: $\Q_{5}$ Degree $d$: $10$ Ramification exponent $e$: $5$ Residue field degree $f$: $2$ Discriminant exponent $c$: $10$ Discriminant root field: $\Q_{5}$ Root number: $1$ $|\Aut(K/\Q_{ 5 })|$: $1$ This field is not Galois 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: $\Q_{5}(\sqrt{2})$ $\cong \Q_{5}(t)$ where $t$ is a root of $$x^{2} + 2$$ Relative Eisenstein polynomial: $$x^{5} + 10 t x + 5$$$\ \in\Q_{5}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_5:D_5.C_4$ (as 10T18) Inertia group: Intransitive group isomorphic to $C_5^2:C_4$ Unramified degree: $2$ Tame degree: $4$ Wild slopes: [5/4, 5/4] Galois mean slope: $123/100$ Galois splitting model: $x^{10} - 20 x^{8} - 40 x^{7} + 370 x^{6} + 120 x^{5} - 3100 x^{4} + 3400 x^{3} + 6815 x^{2} - 16200 x + 9232$