Properties

Label 17.8.6.1
Base \(\Q_{17}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(6\)
Galois group $C_4\times C_2$

Related objects

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Defining polynomial

\( x^{8} - 119 x^{4} + 23409 \)

Invariants

Base field: $\Q_{17}$
Degree $d$ : $8$
Ramification exponent $e$ : $4$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $6$
Discriminant root field: $\Q_{17}$
Root number: $-1$
$|\Gal(K/\Q_{ 17 })|$: $8$
This field is Galois and abelian over $\Q_{17}$.

Intermediate fields

$\Q_{17}(\sqrt{*})$, $\Q_{17}(\sqrt{17})$, $\Q_{17}(\sqrt{17*})$, 17.4.2.1, 17.4.3.2, 17.4.3.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{17}(\sqrt{*})$ $\cong \Q_{17}(t)$ where $t$ is a root of \( x^{2} - x + 3 \)
Relative Eisenstein polynomial:$ x^{4} - 17 t^{4} \in\Q_{17}(t)[x]$

Invariants of the Galois closure

Galois group:$C_4\times C_2$
Inertia group:Intransitive group isomorphic to $C_4$
Unramified degree:$2$
Tame degree:$4$
Wild slopes:None
Galois Mean Slope:$3/4$
Global Splitting Model:Not computed