Defining polynomial
|
\(x^{10} + 235\)
|
Invariants
| Base field: | $\Q_{47}$ |
|
| Degree $d$: | $10$ |
|
| Ramification index $e$: | $10$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $9$ |
|
| Discriminant root field: | $\Q_{47}(\sqrt{47})$ | |
| Root number: | $i$ | |
| $\Aut(K/\Q_{47})$: | $C_2$ | |
| This field is not Galois over $\Q_{47}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $46 = (47 - 1)$ |
|
Intermediate fields
| $\Q_{47}(\sqrt{47})$, 47.1.5.4a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{47}$ |
|
| Relative Eisenstein polynomial: |
\( x^{10} + 235 \)
|
Ramification polygon
| Residual polynomials: | $z^9 + 10 z^8 + 45 z^7 + 26 z^6 + 22 z^5 + 17 z^4 + 22 z^3 + 26 z^2 + 45 z + 10$ |
| Associated inertia: | $4$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $40$ |
| Galois group: | $C_2\times F_5$ (as 10T5) |
| Inertia group: | $C_{10}$ (as 10T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $10$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.9$ |
| Galois splitting model: |
$x^{10} - 47$
|