Properties

Label 3.8.7.2
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

$\Q_{3}(\sqrt{3})$, 3.4.3.2

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{3}$
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:$x^{8} - 2 x^{7} + x^{6} - 5 x^{5} + 7 x^{4} - 2 x^{3} + x^{2} - 5 x + 1$