Properties

Label 17.3.2.1
Base \(\Q_{17}\)
Degree \(3\)
e \(3\)
f \(1\)
c \(2\)
Galois group $S_3$ (as 3T2)

Related objects

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Defining polynomial

\( x^{3} - 17 \)

Invariants

Base field: $\Q_{17}$
Degree $d$ : $3$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $2$
Discriminant root field: $\Q_{17}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 17 })|$: $1$
This field is not Galois over $\Q_{17}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 17 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{17}$
Relative Eisenstein polynomial:\( x^{3} - 17 \)

Invariants of the Galois closure

Galois group:$S_3$ (as 3T2)
Inertia group:$C_3$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{3} - 17$