Properties

Label 2.10.0.1
Base \(\Q_{2}\)
Degree \(10\)
e \(1\)
f \(10\)
c \(0\)
Galois group $C_{10}$ (as 10T1)

Related objects

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Defining polynomial

\( x^{10} - x^{3} + 1 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $10$
Ramification exponent $e$ : $1$
Residue field degree $f$ : $10$
Discriminant exponent $c$ : $0$
Discriminant root field: $\Q_{2}(\sqrt{*})$
Root number: $1$
$|\Gal(K/\Q_{ 2 })|$: $10$
This field is Galois and abelian over $\Q_{2}$.

Intermediate fields

$\Q_{2}(\sqrt{*})$, 2.5.0.1

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

Unramified/totally ramified tower

Unramified subfield:2.10.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{10} - x^{3} + 1 \)
Relative Eisenstein polynomial:$ x - 2 \in\Q_{2}(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:\( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \)