Properties

Label 3.10.9.2
Base \(\Q_{3}\)
Degree \(10\)
e \(10\)
f \(1\)
c \(9\)
Galois group $F_{5}\times C_2$ (as 10T5)

Related objects

Learn more about

Defining polynomial

\(x^{10} + 3\)  Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $10$
Ramification exponent $e$: $10$
Residue field degree $f$: $1$
Discriminant exponent $c$: $9$
Discriminant root field: $\Q_{3}(\sqrt{3\cdot 2})$
Root number: $-i$
$|\Aut(K/\Q_{ 3 })|$: $2$
This field is not Galois over $\Q_{3}.$

Intermediate fields

$\Q_{3}(\sqrt{3\cdot 2})$, 3.5.4.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{3}$
Relative Eisenstein polynomial:\( x^{10} + 3 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_2\times F_5$ (as 10T5)
Inertia group:$C_{10}$
Unramified degree:$4$
Tame degree:$10$
Wild slopes:None
Galois mean slope:$9/10$
Galois splitting model:$x^{10} + 3$  Toggle raw display