Properties

Label 5.6.4.1
Base \(\Q_{5}\)
Degree \(6\)
e \(3\)
f \(2\)
c \(4\)
Galois group $S_3$ (as 6T2)

Related objects

Learn more

Defining polynomial

\(x^{6} + 25 x^{3} + 200\)  Toggle raw display

Invariants

Base field: $\Q_{5}$
Degree $d$: $6$
Ramification exponent $e$: $3$
Residue field degree $f$: $2$
Discriminant exponent $c$: $4$
Discriminant root field: $\Q_{5}(\sqrt{2})$
Root number: $1$
$|\Gal(K/\Q_{ 5 })|$: $6$
This field is Galois over $\Q_{5}.$

Intermediate fields

$\Q_{5}(\sqrt{2})$, 5.3.2.1 x3

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{5}(\sqrt{2})$ $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{2} - x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{3} - 5 t^{3} \)$\ \in\Q_{5}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$S_3$ (as 6T2)
Inertia group:Intransitive group isomorphic to $C_3$
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{6} - 3 x^{5} + x^{4} + 3 x^{3} + x^{2} - 3 x + 1$