Properties

Label 3.8.7.1
Base \(\Q_{3}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(7\)
Galois group $QD_{16}$ (as 8T8)

Related objects

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Defining polynomial

\( x^{8} + 3 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $8$
Ramification exponent $e$ : $8$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $7$
Discriminant root field: $\Q_{3}(\sqrt{3})$
Root number: $i$
$|\Aut(K/\Q_{ 3 })|$: $2$
This field is not Galois over $\Q_{3}$.

Intermediate fields

$\Q_{3}(\sqrt{3*})$, 3.4.3.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{3}$
Relative Eisenstein polynomial:\( x^{8} + 3 \)

Invariants of the Galois closure

Galois group:$SD_{16}$ (as 8T8)
Inertia group:$C_8$
Unramified degree:$2$
Tame degree:$8$
Wild slopes:None
Galois mean slope:$7/8$
Galois splitting model:$x^{8} + 24 x^{6} + 210 x^{4} + 792 x^{2} + 1083$