Properties

Label 5.9.6.1
Base \(\Q_{5}\)
Degree \(9\)
e \(3\)
f \(3\)
c \(6\)
Galois group $S_3\times C_3$ (as 9T4)

Related objects

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Defining polynomial

\( x^{9} - 25 x^{3} + 250 \)

Invariants

Base field: $\Q_{5}$
Degree $d$ : $9$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $3$
Discriminant exponent $c$ : $6$
Discriminant root field: $\Q_{5}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 5 })|$: $3$
This field is not Galois over $\Q_{5}$.

Intermediate fields

5.3.2.1, 5.3.0.1

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

Unramified/totally ramified tower

Unramified subfield:5.3.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{3} - x + 2 \)
Relative Eisenstein polynomial:$ x^{3} - 5 t \in\Q_{5}(t)[x]$

Invariants of the Galois closure

Galois group:$C_3\times S_3$ (as 9T4)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$6$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{9} - 4 x^{8} + 3 x^{7} + 6 x^{6} - 18 x^{5} + 26 x^{4} - 8 x^{3} - 15 x^{2} + 14 x - 13$