Properties

Label 3.5.4.1
Base \(\Q_{3}\)
Degree \(5\)
e \(5\)
f \(1\)
c \(4\)
Galois group $F_5$ (as 5T3)

Related objects

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Defining polynomial

\( x^{5} - 3 \)

Invariants

Base field: $\Q_{3}$
Degree $d$: $5$
Ramification exponent $e$: $5$
Residue field degree $f$: $1$
Discriminant exponent $c$: $4$
Discriminant root field: $\Q_{3}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 3 })|$: $1$
This field is not Galois over $\Q_{3}.$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 3 }$.

Unramified/totally ramified tower

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

Invariants of the Galois closure

Galois group:$F_5$ (as 5T3)
Inertia group:$C_5$
Unramified degree:$4$
Tame degree:$5$
Wild slopes:None
Galois mean slope:$4/5$
Galois splitting model:$x^{5} - 3$