Properties

Label 2.9.8.1
Base \(\Q_{2}\)
Degree \(9\)
e \(9\)
f \(1\)
c \(8\)
Galois group $(C_9:C_3):C_2$ (as 9T10)

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Defining polynomial

\(x^{9} - 2\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $9$
Ramification exponent $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
Discriminant root field: $\Q_{2}$
Root number: $1$
$\card{ \Aut(K/\Q_{ 2 }) }$: $1$
This field is not Galois over $\Q_{2}.$
Visible slopes:None

Intermediate fields

2.3.2.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}$
Relative Eisenstein polynomial: \( x^{9} - 2 \) Copy content Toggle raw display
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$D_9:C_3$ (as 9T10)
Inertia group:$C_9$ (as 9T1)
Wild inertia group:$C_1$
Unramified degree:$6$
Tame degree:$9$
Wild slopes:None
Galois mean slope:$8/9$
Galois splitting model:$x^{9} - 2$