Properties

Label 23.14.0.1
Base \(\Q_{23}\)
Degree \(14\)
e \(1\)
f \(14\)
c \(0\)
Galois group $C_{14}$ (as 14T1)

Related objects

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Defining polynomial

\(x^{14} - x + 7\)  Toggle raw display

Invariants

Base field: $\Q_{23}$
Degree $d$: $14$
Ramification exponent $e$: $1$
Residue field degree $f$: $14$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{23}(\sqrt{5})$
Root number: $1$
$|\Gal(K/\Q_{ 23 })|$: $14$
This field is Galois and abelian over $\Q_{23}.$

Intermediate fields

$\Q_{23}(\sqrt{5})$, 23.7.0.1

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

Unramified/totally ramified tower

Unramified subfield:23.14.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of \( x^{14} - x + 7 \)  Toggle raw display
Relative Eisenstein polynomial:\( x - 23 \)$\ \in\Q_{23}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_{14}$ (as 14T1)
Inertia group:trivial
Wild inertia group:$C_1$
Unramified degree:$14$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{14} - x^{13} + 3 x^{12} - 11 x^{11} + 44 x^{10} + 156 x^{9} - 250 x^{8} - 749 x^{7} + 1560 x^{6} + 4490 x^{5} + 3202 x^{4} + 755 x^{3} + 4050 x^{2} + 1750 x + 625$