Properties

Label 43.7.6.1
Base \(\Q_{43}\)
Degree \(7\)
e \(7\)
f \(1\)
c \(6\)
Galois group $C_7$ (as 7T1)

Related objects

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Defining polynomial

\(x^{7} - 43\)  Toggle raw display

Invariants

Base field: $\Q_{43}$
Degree $d$: $7$
Ramification exponent $e$: $7$
Residue field degree $f$: $1$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{43}$
Root number: $1$
$|\Gal(K/\Q_{ 43 })|$: $7$
This field is Galois and abelian over $\Q_{43}.$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 43 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{43}$
Relative Eisenstein polynomial:\( x^{7} - 43 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_7$ (as 7T1)
Inertia group:$C_7$
Unramified degree:$1$
Tame degree:$7$
Wild slopes:None
Galois mean slope:$6/7$
Galois splitting model:Not computed