Properties

Label 29.4.3.3
Base \(\Q_{29}\)
Degree \(4\)
e \(4\)
f \(1\)
c \(3\)
Galois group $C_4$ (as 4T1)

Related objects

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Defining polynomial

\(x^{4} + 58\)  Toggle raw display

Invariants

Base field: $\Q_{29}$
Degree $d$: $4$
Ramification exponent $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $3$
Discriminant root field: $\Q_{29}(\sqrt{29\cdot 2})$
Root number: $1$
$|\Gal(K/\Q_{ 29 })|$: $4$
This field is Galois and abelian over $\Q_{29}.$

Intermediate fields

$\Q_{29}(\sqrt{29\cdot 2})$

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{29}$
Relative Eisenstein polynomial:\( x^{4} + 58 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_4$ (as 4T1)
Inertia group:$C_4$
Wild inertia group:$C_1$
Unramified degree:$1$
Tame degree:$4$
Wild slopes:None
Galois mean slope:$3/4$
Galois splitting model:$x^{4} + 116 x^{2} + 2842$