Defining polynomial
\(x^{11} + 5\) |
Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $11$ |
Ramification exponent $e$: | $11$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $10$ |
Discriminant root field: | $\Q_{5}$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 5 }) }$: | $1$ |
This field is not Galois over $\Q_{5}.$ | |
Visible slopes: | None |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 5 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{5}$ |
Relative Eisenstein polynomial: | \( x^{11} + 5 \) |
Ramification polygon
Residual polynomials: | $z^{10} + z^{9} + 2z^{5} + 2z^{4} + 1$ |
Associated inertia: | $5$ |
Indices of inseparability: | $[0]$ |