# Properties

 Label 2.4.9.4 Base $$\Q_{2}$$ Degree $$4$$ e $$4$$ f $$1$$ c $$9$$ Galois group $D_{4}$ (as 4T3)

# Related objects

## Defining polynomial

 $$x^{4} + 2 x^{2} + 10$$

## Invariants

 Base field: $\Q_{2}$ Degree $d$ : $4$ Ramification exponent $e$ : $4$ Residue field degree $f$ : $1$ Discriminant exponent $c$ : $9$ Discriminant root field: $\Q_{2}(\sqrt{2*})$ Root number: $-1$ $|\Aut(K/\Q_{ 2 })|$: $2$ This field is not Galois over $\Q_{2}$.

## 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: $\Q_{2}$ Relative Eisenstein polynomial: $$x^{4} + 2 x^{2} + 10$$

## Invariants of the Galois closure

 Galois group: $D_4$ (as 4T3) Inertia group: $D_{4}$ Unramified degree: $1$ Tame degree: $1$ Wild slopes: [2, 3, 7/2] Galois mean slope: $11/4$ Galois splitting model: $x^{4} + 2 x^{2} + 10$
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