Properties

Label 5.15.15.18
Base \(\Q_{5}\)
Degree \(15\)
e \(5\)
f \(3\)
c \(15\)
Galois group $F_5\times C_3$ (as 15T8)

Related objects

Learn more about

Defining polynomial

\( x^{15} + 20 x^{13} + 10 x^{12} + 15 x^{11} + 17 x^{10} + 10 x^{9} + 15 x^{8} + 20 x^{7} + 20 x^{6} + 7 x^{5} + 20 x^{4} + 20 x^{3} + 10 x^{2} + 10 x + 2 \)

Invariants

Base field: $\Q_{5}$
Degree $d$: $15$
Ramification exponent $e$: $5$
Residue field degree $f$: $3$
Discriminant exponent $c$: $15$
Discriminant root field: $\Q_{5}(\sqrt{5})$
Root number: $-1$
$|\Aut(K/\Q_{ 5 })|$: $3$
This field is not Galois over $\Q_{5}.$

Intermediate fields

5.3.0.1, 5.5.5.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:5.3.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{3} - x + 2 \)
Relative Eisenstein polynomial:$ x^{5} + \left(5 t + 10\right) x^{4} + \left(10 t^{2} + 15 t + 20\right) x^{3} + \left(10 t^{2} + 20 t + 10\right) x^{2} + 10 t x + 10 t^{2} + 20 t \in\Q_{5}(t)[x]$

Invariants of the Galois closure

Galois group:$C_3\times F_5$ (as 15T8)
Inertia group:Intransitive group isomorphic to $F_5$
Unramified degree:$3$
Tame degree:$4$
Wild slopes:[5/4]
Galois mean slope:$23/20$
Galois splitting model:$x^{15} + 5 x^{13} - 25 x^{11} - 54 x^{10} - 100 x^{9} + 170 x^{8} + 25 x^{7} - 210 x^{6} + 130 x^{5} - 1150 x^{4} + 3145 x^{3} + 450 x^{2} - 3800 x + 2792$