Properties

Label 29.7.6.3
Base \(\Q_{29}\)
Degree \(7\)
e \(7\)
f \(1\)
c \(6\)
Galois group $C_7$ (as 7T1)

Related objects

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Defining polynomial

\(x^{7} + 928\)  Toggle raw display

Invariants

Base field: $\Q_{29}$
Degree $d$: $7$
Ramification exponent $e$: $7$
Residue field degree $f$: $1$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{29}$
Root number: $1$
$|\Gal(K/\Q_{ 29 })|$: $7$
This field is Galois and abelian over $\Q_{29}.$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 29 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{29}$
Relative Eisenstein polynomial:\( x^{7} + 928 \)  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