Properties

Label 7.8.7.1
Base \(\Q_{7}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(7\)
Galois group $D_{8}$ (as 8T6)

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Defining polynomial

\(x^{8} + 7\) Copy content Toggle raw display

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: $i$
$\card{ \Aut(K/\Q_{ 7 }) }$: $2$
This field is not Galois over $\Q_{7}.$
Visible slopes:None

Intermediate fields

$\Q_{7}(\sqrt{7\cdot 3})$, 7.4.3.1

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: \( x^{8} + 7 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{7} + z^{6} + 1$
Associated inertia:$2$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$D_8$ (as 8T6)
Inertia group:$C_8$ (as 8T1)
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$8$
Wild slopes:None
Galois mean slope:$7/8$
Galois splitting model:$x^{8} - 3 x^{7} + 7 x^{6} - 14 x^{5} + 21 x^{4} - 21 x^{3} + 21 x^{2} - 10 x + 2$