Properties

Label 7.8.6.2
Base \(\Q_{7}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(6\)
Galois group $D_4$ (as 8T4)

Related objects

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Defining polynomial

\( x^{8} - 49 x^{4} + 3969 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $8$
Ramification exponent $e$ : $4$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $6$
Discriminant root field: $\Q_{7}$
Root number: $1$
$|\Gal(K/\Q_{ 7 })|$: $8$
This field is Galois over $\Q_{7}$.

Intermediate fields

$\Q_{7}(\sqrt{*})$, $\Q_{7}(\sqrt{7})$, $\Q_{7}(\sqrt{7*})$, 7.4.2.1, 7.4.3.1 x2, 7.4.3.2 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{7}(\sqrt{*})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} - x + 3 \)
Relative Eisenstein polynomial:$ x^{4} - 7 t^{4} \in\Q_{7}(t)[x]$

Invariants of the Galois closure

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