Defining polynomial
|
\(x^{9} + 3 x + 3\)
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Invariants
| Base field: | $\Q_{3}$ |
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| Degree $d$: | $9$ |
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| Ramification index $e$: | $9$ |
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| Residue field degree $f$: | $1$ |
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| Discriminant exponent $c$: | $9$ |
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| Discriminant root field: | $\Q_{3}(\sqrt{3})$ | |
| Root number: | $i$ | |
| $\Aut(K/\Q_{3})$: | $C_1$ | |
| This field is not Galois over $\Q_{3}.$ | ||
| Visible Artin slopes: | $[\frac{9}{8}, \frac{9}{8}]$ | |
| Visible Swan slopes: | $[\frac{1}{8},\frac{1}{8}]$ | |
| Means: | $\langle\frac{1}{12}, \frac{1}{9}\rangle$ | |
| Rams: | $(\frac{1}{8}, \frac{1}{8})$ | |
| Jump set: | undefined | |
| Roots of unity: | $2 = (3 - 1)$ |
|
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 3 }$. |
Canonical tower
| Unramified subfield: | $\Q_{3}$ |
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| Relative Eisenstein polynomial: |
\( x^{9} + 3 x + 3 \)
|
Ramification polygon
| Residual polynomials: | $z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 1, 0]$ |
Invariants of the Galois closure
| Galois degree: | $144$ |
| Galois group: | $F_9:C_2$ (as 9T19) |
| Inertia group: | $F_9$ (as 9T15) |
| Wild inertia group: | $C_3^2$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $8$ |
| Galois Artin slopes: | $[\frac{9}{8}, \frac{9}{8}]$ |
| Galois Swan slopes: | $[\frac{1}{8},\frac{1}{8}]$ |
| Galois mean slope: | $1.0972222222222223$ |
| Galois splitting model: | $x^{9} - 3 x^{8} + 15 x^{7} - 33 x^{6} + 102 x^{5} - 132 x^{4} + 309 x^{3} - 219 x^{2} + 339 x - 143$ |