Properties

Label 5.5.8.3
Base \(\Q_{5}\)
Degree \(5\)
e \(5\)
f \(1\)
c \(8\)
Galois group $C_5$ (as 5T1)

Related objects

Learn more about

Defining polynomial

\( x^{5} - 5 x^{4} + 30 \)

Invariants

Base field: $\Q_{5}$
Degree $d$ : $5$
Ramification exponent $e$ : $5$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $8$
Discriminant root field: $\Q_{5}$
Root number: $1$
$|\Gal(K/\Q_{ 5 })|$: $5$
This field is Galois and abelian 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:\( x^{5} - 5 x^{4} + 30 \)

Invariants of the Galois closure

Galois group:$C_5$ (as 5T1)
Inertia group:$C_5$
Unramified degree:$1$
Tame degree:$1$
Wild slopes:[2]
Galois mean slope:$8/5$
Galois splitting model:$x^{5} - 110 x^{3} - 605 x^{2} - 990 x - 451$