Defining polynomial
|
\(x^{3} + 74\)
|
Invariants
| Base field: | $\Q_{37}$ |
| Degree $d$: | $3$ |
| Ramification index $e$: | $3$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $2$ |
| Discriminant root field: | $\Q_{37}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{37})$ $=$$\Gal(K/\Q_{37})$: | $C_3$ |
| This field is Galois and abelian over $\Q_{37}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $36 = (37 - 1)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 37 }$. |
Canonical tower
| Unramified subfield: | $\Q_{37}$ |
| Relative Eisenstein polynomial: |
\( x^{3} + 74 \)
|
Ramification polygon
| Residual polynomials: | $z^2 + 3 z + 3$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $3$ |
| Galois group: | $C_3$ (as 3T1) |
| Inertia group: | $C_3$ (as 3T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $1$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.6666666666666666$ |
| Galois splitting model: | $x^{3} - x^{2} - 86 x + 211$ |