Defining polynomial
| \( x^{10} - 72171 \) |
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $10$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $9$ |
| Discriminant root field: | $\Q_{11}(\sqrt{11})$ |
| Root number: | $-i$ |
| $|\Gal(K/\Q_{ 11 })|$: | $10$ |
| This field is Galois and abelian over $\Q_{11}.$ | |
Intermediate fields
| $\Q_{11}(\sqrt{11})$, 11.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{11}$ |
| Relative Eisenstein polynomial: | \( x^{10} - 72171 \) |