Defining polynomial
\(x^{7} + 42 x^{3} + 7\) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $7$ |
Ramification exponent $e$: | $7$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $9$ |
Discriminant root field: | $\Q_{7}(\sqrt{7\cdot 3})$ |
Root number: | $-i$ |
$\card{ \Aut(K/\Q_{ 7 }) }$: | $1$ |
This field is not Galois over $\Q_{7}.$ | |
Visible slopes: | $[3/2]$ |
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: | \( x^{7} + 42 x^{3} + 7 \) |
Ramification polygon
Residual polynomials: | $z^{3} + 3$ |
Associated inertia: | $3$ |
Indices of inseparability: | $[3, 0]$ |