Base \(\Q_{23}\)
Degree \(7\)
e \(7\)
f \(1\)
c \(6\)
Galois group $C_7:C_3$ (as 7T3)

Related objects

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Defining polynomial

\( x^{7} - 23 \)


Base field: $\Q_{23}$
Degree $d$: $7$
Ramification exponent $e$: $7$
Residue field degree $f$: $1$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{23}$
Root number: $1$
$|\Aut(K/\Q_{ 23 })|$: $1$
This field is not Galois over $\Q_{23}.$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 23 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{23}$
Relative Eisenstein polynomial:\( x^{7} - 23 \)

Invariants of the Galois closure

Galois group:$C_7:C_3$ (as 7T3)
Inertia group:$C_7$
Unramified degree:$3$
Tame degree:$7$
Wild slopes:None
Galois mean slope:$6/7$
Galois splitting model:Not computed