Defining polynomial
\( x^{7} + 35 x^{4} + 7 \) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $7$ |
Ramification exponent $e$: | $7$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $10$ |
Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
Root number: | $1$ |
$|\Aut(K/\Q_{ 7 })|$: | $1$ |
This field is not Galois over $\Q_{7}.$ |
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: | \( x^{7} + 35 x^{4} + 7 \) |
Invariants of the Galois closure
Galois group: | $F_7$ (as 7T4) |
Inertia group: | $C_7:C_3$ |
Unramified degree: | $2$ |
Tame degree: | $3$ |
Wild slopes: | [5/3] |
Galois mean slope: | $32/21$ |
Galois splitting model: | $x^{7} - 7 x^{5} + 14 x^{3} - 7 x - 30$ |