Defining polynomial
\( x^{7} + 21 x^{6} + 7 \) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $7$ |
Ramification exponent $e$: | $7$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{7}$ |
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: | \( x^{7} + 21 x^{6} + 7 \) |