Properties

Label 2.9.6.1
Base \(\Q_{2}\)
Degree \(9\)
e \(3\)
f \(3\)
c \(6\)
Galois group $S_3\times C_3$ (as 9T4)

Related objects

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Defining polynomial

\( x^{9} - 4 x^{3} + 8 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $9$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $3$
Discriminant exponent $c$ : $6$
Discriminant root field: $\Q_{2}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 2 })|$: $3$
This field is not Galois over $\Q_{2}$.

Intermediate fields

2.3.2.1, 2.3.0.1

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

Unramified/totally ramified tower

Unramified subfield:2.3.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{3} - x + 1 \)
Relative Eisenstein polynomial:$ x^{3} - 2 t \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_3\times S_3$ (as 9T4)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$6$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{9} - 3 x^{7} - x^{6} + 3 x^{5} + 2 x^{4} + 4 x^{3} - x^{2} - 5 x + 1$