Properties

Label 19.8.7.2
Base \(\Q_{19}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(7\)
Galois group $QD_{16}$ (as 8T8)

Related objects

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Defining polynomial

\( x^{8} - 19 \)

Invariants

Base field: $\Q_{19}$
Degree $d$ : $8$
Ramification exponent $e$ : $8$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $7$
Discriminant root field: $\Q_{19}(\sqrt{19*})$
Root number: $-i$
$|\Aut(K/\Q_{ 19 })|$: $2$
This field is not Galois over $\Q_{19}$.

Intermediate fields

$\Q_{19}(\sqrt{19})$, 19.4.3.2

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{19}$
Relative Eisenstein polynomial:\( x^{8} - 19 \)

Invariants of the Galois closure

Galois group:$SD_{16}$ (as 8T8)
Inertia group:$C_8$
Unramified degree:$2$
Tame degree:$8$
Wild slopes:None
Galois mean slope:$7/8$
Galois splitting model:Not computed