Properties

Label 3.8.6.1
Base \(\Q_{3}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(6\)
Galois group $Q_8$ (as 8T5)

Related objects

Learn more

Defining polynomial

\(x^{8} + 9 x^{4} + 36\)  Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $8$
Ramification exponent $e$: $4$
Residue field degree $f$: $2$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{3}$
Root number: $1$
$|\Gal(K/\Q_{ 3 })|$: $8$
This field is Galois over $\Q_{3}.$

Intermediate fields

$\Q_{3}(\sqrt{2})$, $\Q_{3}(\sqrt{3})$, $\Q_{3}(\sqrt{3\cdot 2})$, 3.4.2.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{3}(\sqrt{2})$ $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{2} - x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{4} - 3 t^{2} \)$\ \in\Q_{3}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$Q_8$ (as 8T5)
Inertia group:Intransitive group isomorphic to $C_4$
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$4$
Wild slopes:None
Galois mean slope:$3/4$
Galois splitting model:$x^{8} + 12 x^{6} + 36 x^{4} + 36 x^{2} + 9$

Additional information

There is a tamely ramified extension of $\Q_p$ with Galois group $Q_8$, the quaternion group, if and only if $p\equiv 3 \pmod4$, and for each such $p$, there is a unique extension. This is the first such example, in that is involves the smallest such odd prime $p$.