Defining polynomial
|
\(x^{10} + 44\)
|
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $10$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $9$ |
| Discriminant root field: | $\Q_{11}(\sqrt{11\cdot 2})$ |
| Root number: | $i$ |
| $\card{ \Gal(K/\Q_{ 11 }) }$: | $10$ |
| This field is Galois and abelian over $\Q_{11}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{11}(\sqrt{11\cdot 2})$, 11.5.4.2 |
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: |
\( x^{10} + 44 \)
|
Ramification polygon
| Residual polynomials: | $z^{9} + 10z^{8} + z^{7} + 10z^{6} + z^{5} + 10z^{4} + z^{3} + 10z^{2} + z + 10$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |