Base \(\Q_{43}\)
Degree \(3\)
e \(3\)
f \(1\)
c \(2\)
Galois group $C_3$ (as 3T1)

Related objects

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Defining polynomial

\(x^{3} + 387\)  Toggle raw display


Base field: $\Q_{43}$
Degree $d$: $3$
Ramification exponent $e$: $3$
Residue field degree $f$: $1$
Discriminant exponent $c$: $2$
Discriminant root field: $\Q_{43}$
Root number: $1$
$|\Gal(K/\Q_{ 43 })|$: $3$
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:\( x^{3} + 387 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_3$ (as 3T1)
Inertia group:$C_3$
Unramified degree:$1$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{3} - 129 x - 559$