Properties

Label 5.5.8.1
Base \(\Q_{5}\)
Degree \(5\)
e \(5\)
f \(1\)
c \(8\)
Galois group $C_5$ (as 5T1)

Related objects

Downloads

Learn more

Defining polynomial

\(x^{5} + 20 x^{4} + 105\) Copy content Toggle raw display

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: $1$
$\card{ \Gal(K/\Q_{ 5 }) }$: $5$
This field is Galois and abelian over $\Q_{5}.$
Visible slopes:$[2]$

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: \( x^{5} + 20 x^{4} + 105 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{4} + 4$
Associated inertia:$1$
Indices of inseparability:$[4, 0]$

Invariants of the Galois closure

Galois group:$C_5$ (as 5T1)
Inertia group:$C_5$ (as 5T1)
Wild inertia group:$C_5$
Unramified degree:$1$
Tame degree:$1$
Wild slopes:$[2]$
Galois mean slope:$8/5$
Galois splitting model:$x^{5} - 110 x^{3} - 55 x^{2} + 660 x + 649$