Defining polynomial
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\(x^{3} + 89\)
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Invariants
| Base field: | $\Q_{89}$ |
| Degree $d$: | $3$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $2$ |
| Discriminant root field: | $\Q_{89}(\sqrt{3})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 89 }) }$: | $1$ |
| This field is not Galois over $\Q_{89}.$ | |
| Visible slopes: | None |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 89 }$. |
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{89}$ |
| Relative Eisenstein polynomial: |
\( x^{3} + 89 \)
|
Ramification polygon
| Residual polynomials: | $z^{2} + 3z + 3$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $S_3$ (as 3T2) |
| Inertia group: | $C_3$ (as 3T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: |
$x^{3} - 9 x^{2} + 27 x - 116$
|