Defining polynomial
\(x^{10} + 22\) |
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: | $-i$ |
$\card{ \Gal(K/\Q_{ 11 }) }$: | $10$ |
This field is Galois and abelian over $\Q_{11}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{11}(\sqrt{11})$, 11.5.4.5 |
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: | \( x^{10} + 22 \) |
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: | $[0]$ |