Properties

Label 23.7.0.1
Base \(\Q_{23}\)
Degree \(7\)
e \(1\)
f \(7\)
c \(0\)
Galois group $C_7$ (as 7T1)

Related objects

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Defining polynomial

\( x^{7} - x + 8 \)

Invariants

Base field: $\Q_{23}$
Degree $d$ : $7$
Ramification exponent $e$ : $1$
Residue field degree $f$ : $7$
Discriminant exponent $c$ : $0$
Discriminant root field: $\Q_{23}$
Root number: $1$
$|\Gal(K/\Q_{ 23 })|$: $7$
This field is Galois and abelian over $\Q_{23}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 23 }$.

Unramified/totally ramified tower

Unramified subfield:23.7.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of \( x^{7} - x + 8 \)
Relative Eisenstein polynomial:$ x - 23 \in\Q_{23}(t)[x]$

Invariants of the Galois closure

Galois group:$C_7$ (as 7T1)
Inertia group:Trivial
Unramified degree:$7$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$