Properties

 Label 3.8.6.2 Base $$\Q_{3}$$ Degree $$8$$ e $$4$$ f $$2$$ c $$6$$ Galois group $D_4$ (as 8T4)

Related objects

Defining polynomial

 $$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$

Invariants

 Base field: $\Q_{3}$ Degree $d$: $8$ Ramification exponent $e$: $4$ Residue field degree $f$: $2$ Discriminant exponent $c$: $6$ Discriminant root field: $\Q_{3}$ Root number: $1$ $|\Gal(K/\Q_{ 3 })|$: $8$ This field is Galois over $\Q_{3}.$

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: $\Q_{3}(\sqrt{2})$ $\cong \Q_{3}(t)$ where $t$ is a root of $$x^{2} - x + 2$$ Relative Eisenstein polynomial: $$x^{4} - 3$$$\ \in\Q_{3}(t)[x]$

Invariants of the Galois closure

 Galois group: $D_4$ (as 8T4) Inertia group: Intransitive group isomorphic to $C_4$ Wild inertia group: $C_1$ Unramified degree: $2$ Tame degree: $4$ Wild slopes: None Galois mean slope: $3/4$ Galois splitting model: $x^{8} - 3 x^{7} + 4 x^{6} - 3 x^{5} + 3 x^{4} - 3 x^{3} + 4 x^{2} - 3 x + 1$