Defining polynomial
| \( x^{10} + 2 x^{2} + 2 x + 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $10$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| 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: | \( x^{10} + 2 x^{2} + 2 x + 2 \) |