Properties

Label 71.10.0.1
Base \(\Q_{71}\)
Degree \(10\)
e \(1\)
f \(10\)
c \(0\)
Galois group $C_{10}$ (as 10T1)

Related objects

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Defining polynomial

\(x^{10} - x + 22\)  Toggle raw display

Invariants

Base field: $\Q_{71}$
Degree $d$: $10$
Ramification exponent $e$: $1$
Residue field degree $f$: $10$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{71}(\sqrt{7})$
Root number: $1$
$|\Gal(K/\Q_{ 71 })|$: $10$
This field is Galois and abelian over $\Q_{71}.$

Intermediate fields

$\Q_{71}(\sqrt{7})$, 71.5.0.1

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

Unramified/totally ramified tower

Unramified subfield:71.10.0.1 $\cong \Q_{71}(t)$ where $t$ is a root of \( x^{10} - x + 22 \)  Toggle raw display
Relative Eisenstein polynomial:\( x - 71 \)$\ \in\Q_{71}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_{10}$ (as 10T1)
Inertia group:trivial
Wild inertia group:$C_1$
Unramified degree:$10$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{10} - x^{9} - 18 x^{8} + 13 x^{7} + 91 x^{6} - 47 x^{5} - 143 x^{4} + 7 x^{3} + 72 x^{2} + 23 x + 1$