Defining polynomial
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\(x^{9} + 3 x^{8} + 3 x^{7} + 3 x^{3} + 3\)
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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 $c$: | $15$ |
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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: | $[\frac{3}{2}, 2]$ | |
| Visible Swan slopes: | $[\frac{1}{2},1]$ | |
| Means: | $\langle\frac{1}{3}, \frac{7}{9}\rangle$ | |
| Rams: | $(\frac{1}{2}, 2)$ | |
| Jump set: | undefined | |
| Roots of unity: | $2 = (3 - 1)$ |
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Intermediate fields
| 3.1.3.3a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{3}$ |
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| Relative Eisenstein polynomial: |
\( x^{9} + 3 x^{8} + 3 x^{7} + 3 x^{3} + 3 \)
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Ramification polygon
| Residual polynomials: | $z^3 + 2$,$2 z^2 + 2$ |
| Associated inertia: | $1$,$2$ |
| Indices of inseparability: | $[7, 3, 0]$ |
Invariants of the Galois closure
| Galois degree: | $108$ |
| Galois group: | $C_3^2:D_6$ (as 9T18) |
| Inertia group: | $C_3^2:S_3$ (as 9T12) |
| Wild inertia group: | $\He_3$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $2$ |
| Galois Artin slopes: | $[\frac{3}{2}, \frac{3}{2}, 2]$ |
| Galois Swan slopes: | $[\frac{1}{2},\frac{1}{2},1]$ |
| Galois mean slope: | $1.7962962962962963$ |
| Galois splitting model: | $x^{9} + 3 x^{7} - 12 x^{6} + 9 x^{5} - 24 x^{4} + 51 x^{3} - 18 x^{2} + 48 x - 34$ |