Defining polynomial
\(x^{10} - 2 x^{5} - 2\)  |
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
| $\Q_{2}(\sqrt{-5})$, 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: | \( x^{10} - 160 x^{9} - 40460 x^{8} - 2808640 x^{7} - 98032360 x^{6} - 998918222 x^{5} - 8072348440 x^{4} + 192345129720 x^{3} - 1170688441400 x^{2} + 3109663379160 x - 3510500485802 \) |