Base \(\Q_{13}\)
Degree \(14\)
e \(1\)
f \(14\)
c \(0\)
Galois group $C_{14}$ (as 14T1)

Related objects

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Defining polynomial

\( x^{14} - x + 2 \)


Base field: $\Q_{13}$
Degree $d$ : $14$
Ramification exponent $e$ : $1$
Residue field degree $f$ : $14$
Discriminant exponent $c$ : $0$
Discriminant root field: $\Q_{13}(\sqrt{*})$
Root number: $1$
$|\Gal(K/\Q_{ 13 })|$: $14$
This field is Galois and abelian over $\Q_{13}$.

Intermediate fields


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

Unramified/totally ramified tower

Unramified subfield: $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{14} - x + 2 \)
Relative Eisenstein polynomial:$ x - 13 \in\Q_{13}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{14}$ (as 14T1)
Inertia group:Trivial
Unramified degree:$14$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:Not computed