Defining polynomial
|
\(x^{10} + 2 x^{8} + 2 x^{7} + 2 x^{5} + 2\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $10$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| Root number: | $i$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[2]$ |
Intermediate fields
| 2.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: |
\( x^{10} + 2 x^{8} + 2 x^{7} + 2 x^{5} + 2 \)
|
Ramification polygon
| Residual polynomials: | $z + 1$,$z^{8} + z^{6} + 1$ |
| Associated inertia: | $1$,$4$ |
| Indices of inseparability: | $[5, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2\wr F_5$ (as 10T29) |
| Inertia group: | $C_2\wr C_5$ (as 10T14) |
| Wild inertia group: | $C_2^5$ |
| Unramified degree: | $4$ |
| Tame degree: | $5$ |
| Wild slopes: | $[8/5, 8/5, 8/5, 8/5, 2]$ |
| Galois mean slope: | $71/40$ |
| Galois splitting model: | $x^{10} + 5 x^{8} + 10 x^{6} - 6 x^{5} + 10 x^{4} + 5 x^{2} + 30 x + 19$ |