Defining polynomial
|
\(x^{10} + 88\)
|
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $10$ |
| Ramification index $e$: | $10$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $9$ |
| Discriminant root field: | $\Q_{11}(\sqrt{11})$ |
| Root number: | $-i$ |
| $\Aut(K/\Q_{11})$ $=$$\Gal(K/\Q_{11})$: | $C_{10}$ |
| This field is Galois and abelian over $\Q_{11}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $10 = (11 - 1)$ |
Intermediate fields
| $\Q_{11}(\sqrt{11})$, 11.1.5.4a1.5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{11}$ |
| Relative Eisenstein polynomial: |
\( x^{10} + 88 \)
|
Ramification polygon
| Residual polynomials: | $z^9 + 10 z^8 + z^7 + 10 z^6 + z^5 + 10 z^4 + z^3 + 10 z^2 + z + 10$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |