Defining polynomial
| \( x^{10} - 3 x^{5} + 18 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $10$ |
| Ramification exponent $e$ : | $5$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $8$ |
| Discriminant root field: | $\Q_{3}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 3 })|$: | $2$ |
| This field is not Galois over $\Q_{3}$. | |
Intermediate fields
| $\Q_{3}(\sqrt{*})$, 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: | $\Q_{3}(\sqrt{*})$ $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{2} - x + 2 \) |
| Relative Eisenstein polynomial: | $ x^{5} - 3 t \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $F_5$ (as 10T4) |
| Inertia group: | Intransitive group isomorphic to $C_5$ |
| Unramified degree: | $4$ |
| Tame degree: | $5$ |
| Wild slopes: | None |
| Galois mean slope: | $4/5$ |
| Galois splitting model: | $x^{10} - 5 x^{9} + 5 x^{8} + 10 x^{7} - 15 x^{6} - 8 x^{5} + 40 x^{3} - 35 x^{2} + 10 x - 4$ |