Properties

Label 5.5.5.4
Base \(\Q_{5}\)
Degree \(5\)
e \(5\)
f \(1\)
c \(5\)
Galois group $F_5$ (as 5T3)

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Defining polynomial

\(x^{5} + 10 x + 5\) Copy content Toggle raw display

Invariants

Base field: $\Q_{5}$
Degree $d$: $5$
Ramification exponent $e$: $5$
Residue field degree $f$: $1$
Discriminant exponent $c$: $5$
Discriminant root field: $\Q_{5}(\sqrt{5\cdot 2})$
Root number: $1$
$\card{ \Aut(K/\Q_{ 5 }) }$: $1$
This field is not Galois over $\Q_{5}.$
Visible slopes:$[5/4]$

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: \( x^{5} + 10 x + 5 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 3$
Associated inertia:$1$
Indices of inseparability:$[1, 0]$

Invariants of the Galois closure

Galois group:$F_5$ (as 5T3)
Inertia group:$F_5$ (as 5T3)
Wild inertia group:$C_5$
Unramified degree:$1$
Tame degree:$4$
Wild slopes:$[5/4]$
Galois mean slope:$23/20$
Galois splitting model:$x^{5} - 40 x^{2} - 90 x - 240$