Defining polynomial
\(x^{23} + 69 x^{8} + 23\) |
Invariants
Base field: | $\Q_{23}$ |
Degree $d$: | $23$ |
Ramification exponent $e$: | $23$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $30$ |
Discriminant root field: | $\Q_{23}$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 23 }) }$: | $1$ |
This field is not Galois over $\Q_{23}.$ | |
Visible slopes: | $[15/11]$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 23 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{23}$ |
Relative Eisenstein polynomial: | \( x^{23} + 69 x^{8} + 23 \) |
Ramification polygon
Residual polynomials: | $z^{2} + 22$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[8, 0]$ |
Invariants of the Galois closure
Galois group: | $C_{23}:C_{11}$ (as 23T3) |
Inertia group: | $C_{23}:C_{11}$ (as 23T3) |
Wild inertia group: | $C_{23}$ |
Unramified degree: | $1$ |
Tame degree: | $11$ |
Wild slopes: | $[15/11]$ |
Galois mean slope: | $340/253$ |
Galois splitting model: | Not computed |