Defining polynomial
|
\(x^{7} + 42 x^{6} + 7\)
|
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $7$ |
| Ramification index $e$: | $7$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{7}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{7})$ $=$$\Gal(K/\Q_{7})$: | $C_7$ |
| This field is Galois and abelian over $\Q_{7}.$ | |
| Visible Artin slopes: | $[2]$ |
| Visible Swan slopes: | $[1]$ |
| Means: | $\langle\frac{6}{7}\rangle$ |
| Rams: | $(1)$ |
| Jump set: | undefined |
| Roots of unity: | $6 = (7 - 1)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 7 }$. |
Canonical tower
| Unramified subfield: | $\Q_{7}$ |
| Relative Eisenstein polynomial: |
\( x^{7} + 42 x^{6} + 7 \)
|
Ramification polygon
| Residual polynomials: | $z^6 + 6$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[6, 0]$ |
Invariants of the Galois closure
| Galois degree: | $7$ |
| Galois group: | $C_7$ (as 7T1) |
| Inertia group: | $C_7$ (as 7T1) |
| Wild inertia group: | $C_7$ |
| Galois unramified degree: | $1$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2]$ |
| Galois Swan slopes: | $[1]$ |
| Galois mean slope: | $1.7142857142857142$ |
| Galois splitting model: | $x^{7} - 21 x^{5} - 21 x^{4} + 91 x^{3} + 112 x^{2} - 84 x - 97$ |