Defining polynomial
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\(x^{7} + 7 x^{6} + 7\)
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Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $7$ |
| Ramification exponent $e$: | $7$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 7 }) }$: | $1$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible slopes: | $[2]$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 7 }$. |
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}$ |
| Relative Eisenstein polynomial: |
\( x^{7} + 7 x^{6} + 7 \)
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Ramification polygon
| Residual polynomials: | $z^{6} + 1$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[6, 0]$ |
Invariants of the Galois closure
| Galois group: | $D_7$ (as 7T2) |
| Inertia group: | $C_7$ (as 7T1) |
| Wild inertia group: | $C_7$ |
| Unramified degree: | $2$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2]$ |
| Galois mean slope: | $12/7$ |
| Galois splitting model: |
$x^{7} - 3157 x^{5} - 85526 x^{4} - 49938 x^{3} + 18478208 x^{2} + 141589728 x - 96857088$
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