Properties

Label 23.8.7.1
Base \(\Q_{23}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(7\)
Galois group $D_{8}$ (as 8T6)

Related objects

Learn more about

Defining polynomial

\( x^{8} + 46 \)

Invariants

Base field: $\Q_{23}$
Degree $d$ : $8$
Ramification exponent $e$ : $8$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $7$
Discriminant root field: $\Q_{23}(\sqrt{23})$
Root number: $i$
$|\Aut(K/\Q_{ 23 })|$: $2$
This field is not Galois over $\Q_{23}$.

Intermediate fields

$\Q_{23}(\sqrt{23*})$, 23.4.3.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{23}$
Relative Eisenstein polynomial:\( x^{8} + 46 \)

Invariants of the Galois closure

Galois group:$D_8$ (as 8T6)
Inertia group:$C_8$
Unramified degree:$2$
Tame degree:$8$
Wild slopes:None
Galois mean slope:$7/8$
Galois splitting model:Not computed