Defining polynomial
| \( x^{10} - 18 x^{6} + 81 x^{2} - 243 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $10$ |
| Ramification exponent $e$ : | $2$ |
| Residue field degree $f$ : | $5$ |
| Discriminant exponent $c$ : | $5$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
| Root number: | $-i$ |
| $|\Gal(K/\Q_{ 3 })|$: | $10$ |
| This field is Galois and abelian over $\Q_{3}$. | |
Intermediate fields
| $\Q_{3}(\sqrt{3})$, 3.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 3.5.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{5} - x + 1 \) |
| Relative Eisenstein polynomial: | $ x^{2} - 3 t^{2} \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_{10}$ (as 10T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Unramified degree: | $5$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | $x^{10} - x^{9} - 200 x^{8} + 194 x^{7} + 11877 x^{6} - 3991 x^{5} - 248665 x^{4} - 32090 x^{3} + 1913681 x^{2} + 788162 x - 3646277$ |