Properties

Label 41.14.0.1
Base \(\Q_{41}\)
Degree \(14\)
e \(1\)
f \(14\)
c \(0\)
Galois group $C_{14}$ (as 14T1)

Related objects

Learn more about

Defining polynomial

\( x^{14} - x + 13 \)

Invariants

Base field: $\Q_{41}$
Degree $d$: $14$
Ramification exponent $e$: $1$
Residue field degree $f$: $14$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{41}(\sqrt{*})$
Root number: $1$
$|\Gal(K/\Q_{ 41 })|$: $14$
This field is Galois and abelian over $\Q_{41}.$

Intermediate fields

$\Q_{41}(\sqrt{*})$, 41.7.0.1

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

Unramified/totally ramified tower

Unramified subfield:41.14.0.1 $\cong \Q_{41}(t)$ where $t$ is a root of \( x^{14} - x + 13 \)
Relative Eisenstein polynomial:$ x - 41 \in\Q_{41}(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