Properties

Label 2.9.0.1
Base \(\Q_{2}\)
Degree \(9\)
e \(1\)
f \(9\)
c \(0\)
Galois group $C_9$ (as 9T1)

Related objects

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Defining polynomial

\( x^{9} + x^{4} + 1 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $9$
Ramification exponent $e$ : $1$
Residue field degree $f$ : $9$
Discriminant exponent $c$ : $0$
Discriminant root field: $\Q_{2}$
Root number: $1$
$|\Gal(K/\Q_{ 2 })|$: $9$
This field is Galois and abelian over $\Q_{2}$.

Intermediate fields

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:2.9.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{9} + x^{4} + 1 \)
Relative Eisenstein polynomial:$ x - 2 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_9$ (as 9T1)
Inertia group:Trivial
Unramified degree:$9$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{9} - x^{8} - 8 x^{7} + 7 x^{6} + 21 x^{5} - 15 x^{4} - 20 x^{3} + 10 x^{2} + 5 x - 1$