Properties

Label 7.8.7.2
Base \(\Q_{7}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(7\)
Galois group $D_{8}$ (as 8T6)

Related objects

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Defining polynomial

\( x^{8} - 7 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $8$
Ramification exponent $e$ : $8$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $7$
Discriminant root field: $\Q_{7}(\sqrt{7*})$
Root number: $-i$
$|\Aut(K/\Q_{ 7 })|$: $2$
This field is not Galois over $\Q_{7}$.

Intermediate fields

$\Q_{7}(\sqrt{7})$, 7.4.3.2

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{7}$
Relative Eisenstein polynomial:\( x^{8} - 7 \)

Invariants of the Galois closure

Galois group:$D_8$ (as 8T6)
Inertia group:$C_8$
Unramified degree:$2$
Tame degree:$8$
Wild slopes:None
Galois mean slope:$7/8$
Galois splitting model:$x^{8} - 2 x^{7} + 7 x^{6} - 14 x^{5} + 21 x^{4} - 28 x^{3} + 14 x^{2} - 16 x + 4$