Properties

Label 7.7.8.2
Base \(\Q_{7}\)
Degree \(7\)
e \(7\)
f \(1\)
c \(8\)
Galois group $C_7:C_3$ (as 7T3)

Related objects

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Defining polynomial

\( x^{7} + 7 x^{2} + 7 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $7$
Ramification exponent $e$ : $7$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $8$
Discriminant root field: $\Q_{7}$
Root number: $1$
$|\Aut(K/\Q_{ 7 })|$: $1$
This field is not Galois over $\Q_{7}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 7 }$.

Unramified/totally ramified tower

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

Invariants of the Galois closure

Galois group:$C_7:C_3$ (as 7T3)
Inertia group:$C_7:C_3$
Unramified degree:$1$
Tame degree:$3$
Wild slopes:[4/3]
Galois mean slope:$26/21$
Galois splitting model:$x^{7} - 231 x^{5} - 1386 x^{4} + 4851 x^{3} + 60522 x^{2} + 162701 x + 127941$