Properties

Label 5.5.8.7
Base \(\Q_{5}\)
Degree \(5\)
e \(5\)
f \(1\)
c \(8\)
Galois group $F_5$ (as 5T3)

Related objects

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Defining polynomial

\( x^{5} + 10 x^{4} + 5 \)

Invariants

Base field: $\Q_{5}$
Degree $d$ : $5$
Ramification exponent $e$ : $5$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $8$
Discriminant root field: $\Q_{5}(\sqrt{*})$
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:\( x^{5} + 10 x^{4} + 5 \)

Invariants of the Galois closure

Galois group:$F_5$ (as 5T3)
Inertia group:$C_5$
Unramified degree:$4$
Tame degree:$1$
Wild slopes:[2]
Galois mean slope:$8/5$
Galois splitting model:$x^{5} - 10 x^{3} - 180 x^{2} + 935 x - 1156$