Properties

Label 17.12.0.1
Base \(\Q_{17}\)
Degree \(12\)
e \(1\)
f \(12\)
c \(0\)
Galois group $C_{12}$ (as 12T1)

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Defining polynomial

\(x^{12} + 3 x^{2} - 2 x + 5\)  Toggle raw display

Invariants

Base field: $\Q_{17}$
Degree $d$: $12$
Ramification exponent $e$: $1$
Residue field degree $f$: $12$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{17}(\sqrt{3})$
Root number: $1$
$|\Gal(K/\Q_{ 17 })|$: $12$
This field is Galois and abelian over $\Q_{17}.$

Intermediate fields

$\Q_{17}(\sqrt{3})$, 17.3.0.1, 17.4.0.1, 17.6.0.1

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

Unramified/totally ramified tower

Unramified subfield:17.12.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of \( x^{12} + 3 x^{2} - 2 x + 5 \)  Toggle raw display
Relative Eisenstein polynomial:\( x - 17 \)$\ \in\Q_{17}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_{12}$ (as 12T1)
Inertia group:trivial
Unramified degree:$12$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{12} - x^{11} + 2 x^{10} + 20 x^{9} - 13 x^{8} + 19 x^{7} + 85 x^{6} - 51 x^{5} + 94 x^{4} + 2 x^{3} - 13 x^{2} + 77 x + 47$