Defining polynomial
\(x^{7} + 3\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $7$ |
Ramification exponent $e$: | $7$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
This field is not Galois over $\Q_{3}.$ | |
Visible slopes: | None |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 3 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{3}$ |
Relative Eisenstein polynomial: | \( x^{7} + 3 \) |
Ramification polygon
Residual polynomials: | $z^{6} + z^{5} + 2z^{3} + 2z^{2} + 1$ |
Associated inertia: | $6$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $F_7$ (as 7T4) |
Inertia group: | $C_7$ (as 7T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $6$ |
Tame degree: | $7$ |
Wild slopes: | None |
Galois mean slope: | $6/7$ |
Galois splitting model: | $x^{7} - 3$ |