Defining polynomial
\(x^{11} + 10 x + 9\) |
Invariants
Base field: | $\Q_{11}$ |
Degree $d$: | $11$ |
Ramification exponent $e$: | $1$ |
Residue field degree $f$: | $11$ |
Discriminant exponent $c$: | $0$ |
Discriminant root field: | $\Q_{11}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 11 }) }$: | $11$ |
This field is Galois and abelian over $\Q_{11}.$ | |
Visible slopes: | None |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 11 }$. |
Unramified/totally ramified tower
Unramified subfield: | 11.11.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{11} + 10 x + 9 \) |
Relative Eisenstein polynomial: | \( x - 11 \) $\ \in\Q_{11}(t)[x]$ |
Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
Galois group: | $C_{11}$ (as 11T1) |
Inertia group: | trivial |
Wild inertia group: | $C_1$ |
Unramified degree: | $11$ |
Tame degree: | $1$ |
Wild slopes: | None |
Galois mean slope: | $0$ |
Galois splitting model: | Not computed |