Properties

Label 7.8.0.1
Base \(\Q_{7}\)
Degree \(8\)
e \(1\)
f \(8\)
c \(0\)
Galois group $C_8$

Related objects

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Defining polynomial

\( x^{8} - x + 3 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $8$
Ramification exponent $e$ : $1$
Residue field degree $f$ : $8$
Discriminant exponent $c$ : $0$
Discriminant root field: $\Q_{7}(\sqrt{*})$
Root number: $1$
$|\Gal(K/\Q_{ 7 })|$: $8$
This field is Galois and abelian over $\Q_{7}$.

Intermediate fields

$\Q_{7}(\sqrt{*})$, 7.4.0.1

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

Unramified/totally ramified tower

Unramified subfield:7.8.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{8} - x + 3 \)
Relative Eisenstein polynomial:$ x - 7 \in\Q_{7}(t)[x]$

Invariants of the Galois closure

Galois group:$C_8$
Inertia group:Trivial
Unramified degree:$8$
Tame degree:$1$
Wild slopes:None
Galois Mean Slope:$0$
Global Splitting Model:\( x^{8} - x^{7} - 7 x^{6} + 6 x^{5} + 15 x^{4} - 10 x^{3} - 10 x^{2} + 4 x + 1 \)