Defining polynomial
\(x^{3} + 21\) |
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: | $1$ |
$\card{ \Gal(K/\Q_{ 7 }) }$: | $3$ |
This field is Galois and abelian over $\Q_{7}.$ | |
Visible slopes: | None |
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: | \( x^{3} + 21 \) |
Ramification polygon
Data not computedInvariants of the Galois closure
Galois group: | $C_3$ (as 3T1) |
Inertia group: | $C_3$ (as 3T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $1$ |
Tame degree: | $3$ |
Wild slopes: | None |
Galois mean slope: | $2/3$ |
Galois splitting model: | $x^{3} - 21 x - 28$ |