Base \(\Q_{7}\)
Degree \(9\)
e \(9\)
f \(1\)
c \(8\)
Galois group $C_9:C_3$ (as 9T6)

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Defining polynomial

\(x^{9} + 7\) Copy content Toggle raw display


Base field: $\Q_{7}$
Degree $d$: $9$
Ramification exponent $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
Discriminant root field: $\Q_{7}$
Root number: $1$
$\card{ \Aut(K/\Q_{ 7 }) }$: $3$
This field is not Galois over $\Q_{7}.$
Visible slopes:None

Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{7}$
Relative Eisenstein polynomial: \( x^{9} + 7 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{8} + 2z^{7} + z^{6} + z + 2$
Associated inertia:$3$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_9:C_3$ (as 9T6)
Inertia group:$C_9$ (as 9T1)
Wild inertia group:$C_1$
Unramified degree:$3$
Tame degree:$9$
Wild slopes:None
Galois mean slope:$8/9$
Galois splitting model:$x^{9} - 14 x^{7} + 63 x^{5} - 98 x^{3} + 42 x - 7$