Properties

Label 11.10.9.1
Base \(\Q_{11}\)
Degree \(10\)
e \(10\)
f \(1\)
c \(9\)
Galois group $C_{10}$ (as 10T1)

Related objects

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Defining polynomial

\( x^{10} - 11 \)

Invariants

Base field: $\Q_{11}$
Degree $d$ : $10$
Ramification exponent $e$ : $10$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $9$
Discriminant root field: $\Q_{11}(\sqrt{11})$
Root number: $-i$
$|\Gal(K/\Q_{ 11 })|$: $10$
This field is Galois and abelian over $\Q_{11}$.

Intermediate fields

$\Q_{11}(\sqrt{11})$, 11.5.4.4

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{11}$
Relative Eisenstein polynomial:\( x^{10} - 11 \)

Invariants of the Galois closure

Galois group:$C_{10}$ (as 10T1)
Inertia group:$C_{10}$
Unramified degree:$1$
Tame degree:$10$
Wild slopes:None
Galois mean slope:$9/10$
Galois splitting model:$x^{10} - x^{9} - 10 x^{8} + 10 x^{7} + 34 x^{6} - 34 x^{5} - 43 x^{4} + 43 x^{3} + 12 x^{2} - 12 x + 1$