Defining polynomial
\(x^{5} + 355\) |
Invariants
Base field: | $\Q_{71}$ |
Degree $d$: | $5$ |
Ramification exponent $e$: | $5$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $4$ |
Discriminant root field: | $\Q_{71}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 71 }) }$: | $5$ |
This field is Galois and abelian over $\Q_{71}.$ | |
Visible slopes: | None |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 71 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{71}$ |
Relative Eisenstein polynomial: | \( x^{5} + 355 \) |
Ramification polygon
Residual polynomials: | $z^{4} + 5z^{3} + 10z^{2} + 10z + 5$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |