Properties

Label 7.15.14.1
Base \(\Q_{7}\)
Degree \(15\)
e \(15\)
f \(1\)
c \(14\)
Galois group $F_5\times C_3$ (as 15T8)

Related objects

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Defining polynomial

\( x^{15} - 7 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $15$
Ramification exponent $e$ : $15$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $14$
Discriminant root field: $\Q_{7}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 7 })|$: $3$
This field is not Galois over $\Q_{7}$.

Intermediate fields

7.3.2.2, 7.5.4.1

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:\( x^{15} - 7 \)

Invariants of the Galois closure

Galois group:$C_3\times F_5$ (as 15T8)
Inertia group:$C_{15}$
Unramified degree:$4$
Tame degree:$15$
Wild slopes:None
Galois mean slope:$14/15$
Galois splitting model:Not computed