Defining polynomial
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\(x^{7} + 28 x^{4} + 7\)
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Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $7$ |
| Ramification exponent $e$: | $7$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{7}$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 7 }) }$: | $1$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible slopes: | $[5/3]$ |
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: |
\( x^{7} + 28 x^{4} + 7 \)
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Ramification polygon
| Residual polynomials: | $z^{2} + 5$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[4, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_7:C_3$ (as 7T3) |
| Inertia group: | $C_7:C_3$ (as 7T3) |
| Wild inertia group: | $C_7$ |
| Unramified degree: | $1$ |
| Tame degree: | $3$ |
| Wild slopes: | $[5/3]$ |
| Galois mean slope: | $32/21$ |
| Galois splitting model: |
$x^{7} - 777 x^{5} - 2331 x^{4} + 117327 x^{3} + 318570 x^{2} - 3217816 x - 8294808$
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