Properties

Label 7.7.9.1
Base \(\Q_{7}\)
Degree \(7\)
e \(7\)
f \(1\)
c \(9\)
Galois group $D_{7}$ (as 7T2)

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Defining polynomial

\(x^{7} + 14 x^{3} + 7\) Copy content Toggle raw display

Invariants

Base field: $\Q_{7}$
Degree $d$: $7$
Ramification exponent $e$: $7$
Residue field degree $f$: $1$
Discriminant exponent $c$: $9$
Discriminant root field: $\Q_{7}(\sqrt{7})$
Root number: $i$
$\card{ \Aut(K/\Q_{ 7 }) }$: $1$
This field is not Galois over $\Q_{7}.$
Visible slopes:$[3/2]$

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: \( x^{7} + 14 x^{3} + 7 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{3} + 1$
Associated inertia:$1$
Indices of inseparability:$[3, 0]$

Invariants of the Galois closure

Galois group:$D_7$ (as 7T2)
Inertia group:$D_7$ (as 7T2)
Wild inertia group:$C_7$
Unramified degree:$1$
Tame degree:$2$
Wild slopes:$[3/2]$
Galois mean slope:$19/14$
Galois splitting model: $x^{7} - 182 x^{5} - 182 x^{4} + 5369 x^{3} + 15834 x^{2} - 25168$ Copy content Toggle raw display