Properties

Label 2.7.6.1
Base \(\Q_{2}\)
Degree \(7\)
e \(7\)
f \(1\)
c \(6\)
Galois group $C_7:C_3$ (as 7T3)

Related objects

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Defining polynomial

\( x^{7} - 2 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $7$
Ramification exponent $e$ : $7$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $6$
Discriminant root field: $\Q_{2}$
Root number: $1$
$|\Aut(K/\Q_{ 2 })|$: $1$
This field is not Galois over $\Q_{2}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 2 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}$
Relative Eisenstein polynomial:\( x^{7} - 2 \)

Invariants of the Galois closure

Galois group:$C_7:C_3$ (as 7T3)
Inertia group:$C_7$
Unramified degree:$3$
Tame degree:$7$
Wild slopes:None
Galois mean slope:$6/7$
Galois splitting model:$x^{7} - 14 x^{5} + 56 x^{3} - 56 x - 22$