Defining polynomial
\( x^{3} - 7 \) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $3$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $2$ |
Discriminant root field: | $\Q_{7}$ |
Root number: | $1$ |
$|\Gal(K/\Q_{ 7 })|$: | $3$ |
This field is Galois and abelian 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: | \( x^{3} - 7 \) |
Invariants of the Galois closure
Galois group: | $C_3$ (as 3T1) |
Inertia group: | $C_3$ |
Unramified degree: | $1$ |
Tame degree: | $3$ |
Wild slopes: | None |
Galois mean slope: | $2/3$ |
Galois splitting model: | $x^{3} - x^{2} - 2 x + 1$ |