Defining polynomial
\(x^{8} + 9\) |
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: | $1$ |
$\card{ \Gal(K/\Q_{ 3 }) }$: | $8$ |
This field is Galois over $\Q_{3}.$ | |
Visible slopes: | None |
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 \( x^{2} + 2 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{4} + 3 t + 3 \) $\ \in\Q_{3}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{3} + z^{2} + 1$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
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 $p$.