Defining polynomial
\(x^{23} + 4 x + 64\)
|
Invariants
Base field: | $\Q_{71}$ |
Degree $d$: | $23$ |
Ramification exponent $e$: | $1$ |
Residue field degree $f$: | $23$ |
Discriminant exponent $c$: | $0$ |
Discriminant root field: | $\Q_{71}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 71 }) }$: | $23$ |
This field is Galois and abelian over $\Q_{71}.$ | |
Visible slopes: | None |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 71 }$. |
Unramified/totally ramified tower
Unramified subfield: | 71.23.0.1 $\cong \Q_{71}(t)$ where $t$ is a root of
\( x^{23} + 4 x + 64 \)
|
Relative Eisenstein polynomial: |
\( x - 71 \)
$\ \in\Q_{71}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
Galois group: | $C_{23}$ (as 23T1) |
Inertia group: | trivial |
Wild inertia group: | $C_1$ |
Unramified degree: | $23$ |
Tame degree: | $1$ |
Wild slopes: | None |
Galois mean slope: | $0$ |
Galois splitting model: | Not computed |