Defining polynomial
\(x^{10} + 2 x^{8} - 2 x^{7} + 2 x^{5} + 2 x^{2} + 6\)  |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $10$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-5})$ |
| Root number: | $i$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
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: | \( x^{10} + 2 x^{8} - 2 x^{7} + 2 x^{5} + 2 x^{2} + 6 \) |
Invariants of the Galois closure
| Galois group: | $((C_2^4 : C_5):C_4)\times C_2$ (as 10T29) |
| Inertia group: | $C_2 \times (C_2^4 : C_5)$ |
| 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: | $x^{10} + 5 x^{8} - 6 x^{7} + 4 x^{6} - 4 x^{5} + 30 x^{4} - 42 x^{3} + 39 x^{2} - 36 x + 23$ |