Defining polynomial
|
\(x^{3} + 111\)
|
Invariants
| Base field: | $\Q_{37}$ |
| Degree $d$: | $3$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $2$ |
| Discriminant root field: | $\Q_{37}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 37 }) }$: | $3$ |
| This field is Galois and abelian over $\Q_{37}.$ | |
| Visible slopes: | None |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 37 }$. |
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{37}$ |
| Relative Eisenstein polynomial: |
\( x^{3} + 111 \)
|
Ramification polygon
| Residual polynomials: | $z^{2} + 3z + 3$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |