Properties

Label 3.7.6.1
Base \(\Q_{3}\)
Degree \(7\)
e \(7\)
f \(1\)
c \(6\)
Galois group $F_7$ (as 7T4)

Related objects

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Defining polynomial

\( x^{7} - 3 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $7$
Ramification exponent $e$ : $7$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $6$
Discriminant root field: $\Q_{3}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 3 })|$: $1$
This field is not Galois over $\Q_{3}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 3 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{3}$
Relative Eisenstein polynomial:\( x^{7} - 3 \)

Invariants of the Galois closure

Galois group:$F_7$ (as 7T4)
Inertia group:$C_7$
Unramified degree:$6$
Tame degree:$7$
Wild slopes:None
Galois mean slope:$6/7$
Galois splitting model:$x^{7} - 3$