Defining polynomial
|
\(x^{10} + 13\)
|
Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $10$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $9$ |
| Discriminant root field: | $\Q_{13}(\sqrt{13})$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 13 }) }$: | $2$ |
| This field is not Galois over $\Q_{13}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{13}(\sqrt{13})$, 13.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{13}$ |
| Relative Eisenstein polynomial: |
\( x^{10} + 13 \)
|
Ramification polygon
Not computedInvariants of the Galois closure
| Galois group: | $C_2\times F_5$ (as 10T5) |
| Inertia group: | $C_{10}$ (as 10T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $4$ |
| Tame degree: | $10$ |
| Wild slopes: | None |
| Galois mean slope: | $9/10$ |
| Galois splitting model: | $x^{10} - 13$ |