Properties

Label 11.10.0.1
Base \(\Q_{11}\)
Degree \(10\)
e \(1\)
f \(10\)
c \(0\)
Galois group $C_{10}$ (as 10T1)

Related objects

Learn more about

Defining polynomial

\( x^{10} + x^{2} - x + 6 \)

Invariants

Base field: $\Q_{11}$
Degree $d$ : $10$
Ramification exponent $e$ : $1$
Residue field degree $f$ : $10$
Discriminant exponent $c$ : $0$
Discriminant root field: $\Q_{11}(\sqrt{*})$
Root number: $1$
$|\Gal(K/\Q_{ 11 })|$: $10$
This field is Galois and abelian over $\Q_{11}$.

Intermediate fields

$\Q_{11}(\sqrt{*})$, 11.5.0.1

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

Unramified/totally ramified tower

Unramified subfield:11.10.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{10} + x^{2} - x + 6 \)
Relative Eisenstein polynomial:$ x - 11 \in\Q_{11}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{10}$ (as 10T1)
Inertia group:Trivial
Unramified degree:$10$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{10} - x^{9} + 2 x^{8} + 16 x^{7} - 9 x^{6} + 11 x^{5} + 43 x^{4} - 6 x^{3} + 63 x^{2} - 20 x + 25$