Properties

Label 31.10.0.1
Base \(\Q_{31}\)
Degree \(10\)
e \(1\)
f \(10\)
c \(0\)
Galois group $C_{10}$ (as 10T1)

Related objects

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Defining polynomial

\(x^{10} - x + 11\)  Toggle raw display

Invariants

Base field: $\Q_{31}$
Degree $d$: $10$
Ramification exponent $e$: $1$
Residue field degree $f$: $10$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{31}(\sqrt{3})$
Root number: $1$
$|\Gal(K/\Q_{ 31 })|$: $10$
This field is Galois and abelian over $\Q_{31}.$

Intermediate fields

$\Q_{31}(\sqrt{3})$, 31.5.0.1

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

Unramified/totally ramified tower

Unramified subfield:31.10.0.1 $\cong \Q_{31}(t)$ where $t$ is a root of \( x^{10} - x + 11 \)  Toggle raw display
Relative Eisenstein polynomial:\( x - 31 \)$\ \in\Q_{31}(t)[x]$  Toggle raw display

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:$x^{10} - x^{9} - 27 x^{8} + 56 x^{7} + 161 x^{6} - 500 x^{5} + x^{4} + 1023 x^{3} - 916 x^{2} + 202 x - 13$