Defining polynomial
\(x^{7} + 42 x^{2} + 7\) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $7$ |
Ramification exponent $e$: | $7$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $8$ |
Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 7 }) }$: | $1$ |
This field is not Galois over $\Q_{7}.$ | |
Visible slopes: | $[4/3]$ |
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: | \( x^{7} + 42 x^{2} + 7 \) |
Ramification polygon
Residual polynomials: | $z^{2} + 2$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[2, 0]$ |