Properties

Label 5.3.2.1
Base \(\Q_{5}\)
Degree \(3\)
e \(3\)
f \(1\)
c \(2\)
Galois group $S_3$ (as 3T2)

Related objects

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Defining polynomial

\(x^{3} - 5\)  Toggle raw display

Invariants

Base field: $\Q_{5}$
Degree $d$: $3$
Ramification exponent $e$: $3$
Residue field degree $f$: $1$
Discriminant exponent $c$: $2$
Discriminant root field: $\Q_{5}(\sqrt{2})$
Root number: $1$
$|\Aut(K/\Q_{ 5 })|$: $1$
This field is not Galois over $\Q_{5}.$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 5 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{5}$
Relative Eisenstein polynomial:\( x^{3} - 5 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$S_3$ (as 3T2)
Inertia group:$C_3$
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{3} - 5$