Defining polynomial
\(x^{7} + 2\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $7$ |
Ramification exponent $e$: | $7$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{2}$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $1$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | None |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 2 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{2}$ |
Relative Eisenstein polynomial: | \( x^{7} + 2 \) |
Ramification polygon
Residual polynomials: | $z^{6} + z^{5} + z^{4} + z^{3} + z^{2} + z + 1$ |
Associated inertia: | $3$ |
Indices of inseparability: | $[0]$ |