Defining polynomial
\(x^{10} - 198 x^{5} - 10043\) |
Invariants
Base field: | $\Q_{11}$ |
Degree $d$: | $10$ |
Ramification exponent $e$: | $5$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $8$ |
Discriminant root field: | $\Q_{11}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 11 }) }$: | $10$ |
This field is Galois and abelian over $\Q_{11}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{11}(\sqrt{2})$, 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: | $\Q_{11}(\sqrt{2})$ $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{2} + 7 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{5} + 44 t + 55 \) $\ \in\Q_{11}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{4} + 5z^{3} + 10z^{2} + 10z + 5$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_{10}$ (as 10T1) |
Inertia group: | Intransitive group isomorphic to $C_5$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $5$ |
Wild slopes: | None |
Galois mean slope: | $4/5$ |
Galois splitting model: | $x^{10} - x^{9} + 2 x^{8} + 326 x^{7} - 536 x^{6} + 4816 x^{5} + 43381 x^{4} - 339673 x^{3} + 908642 x^{2} - 1781528 x + 1928783$ |