Error: Incorrect local algebra for Q(zeta3)
Defining polynomial
$( x^{2} + 2 x + 2 )^{3} + 6 ( x^{2} + 2 x + 2 ) + 3$
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Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $6$ |
Ramification index $e$: | $3$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
Root number: | $1$ |
$\Aut(K/\Q_{3})$: | $C_2$ |
This field is not Galois over $\Q_{3}.$ | |
Visible Artin slopes: | $[\frac{3}{2}]$ |
Visible Swan slopes: | $[\frac{1}{2}]$ |
Means: | $\langle\frac{1}{3}\rangle$ |
Rams: | $(\frac{1}{2})$ |
Jump set: | undefined |
Roots of unity: | $8 = (3^{ 2 } - 1)$ |
Intermediate fields
$\Q_{3}(\sqrt{2})$, 3.1.3.3a1.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
Unramified subfield: | $\Q_{3}(\sqrt{2})$ $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{2} + 2 x + 2 \)
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Relative Eisenstein polynomial: |
\( x^{3} + 6 x + 3 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
Residual polynomials: | $z + (2 t + 2)$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
Galois degree: | $12$ |
Galois group: | $D_6$ (as 6T3) |
Inertia group: | Intransitive group isomorphic to $S_3$ |
Wild inertia group: | $C_3$ |
Galois unramified degree: | $2$ |
Galois tame degree: | $2$ |
Galois Artin slopes: | $[\frac{3}{2}]$ |
Galois Swan slopes: | $[\frac{1}{2}]$ |
Galois mean slope: | $1.1666666666666667$ |
Galois splitting model: | $x^{6} - 3 x^{5} + 21 x^{4} - 31 x^{3} + 63 x^{2} - 45 x + 50$ |