Defining polynomial
| \( x^{10} - 899 x^{5} + 204363 \) |
Invariants
| Base field: | $\Q_{29}$ |
| Degree $d$ : | $10$ |
| Ramification exponent $e$ : | $5$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $8$ |
| Discriminant root field: | $\Q_{29}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 29 })|$: | $10$ |
| This field is Galois over $\Q_{29}$. | |
Intermediate fields
| $\Q_{29}(\sqrt{*})$, 29.5.4.1 x5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{29}(\sqrt{*})$ $\cong \Q_{29}(t)$ where $t$ is a root of \( x^{2} - x + 3 \) |
| Relative Eisenstein polynomial: | $ x^{5} - 29 t^{5} \in\Q_{29}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $D_5$ (as 10T2) |
| Inertia group: | Intransitive group isomorphic to $C_5$ |
| Unramified degree: | $2$ |
| Tame degree: | $5$ |
| Wild slopes: | None |
| Galois mean slope: | $4/5$ |
| Galois splitting model: | Not computed |