Defining polynomial
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$( x^{4} + 2 x^{3} + 2 )^{4} + 3$
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Invariants
| Base field: | $\Q_{3}$ |
|
| Degree $d$: | $16$ |
|
| Ramification index $e$: | $4$ |
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| Residue field degree $f$: | $4$ |
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| Discriminant exponent $c$: | $12$ |
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| Discriminant root field: | $\Q_{3}$ | |
| Root number: | $-1$ | |
| $\Aut(K/\Q_{3})$ $=$ $\Gal(K/\Q_{3})$: | $C_4:C_4$ | |
| This field is Galois over $\Q_{3}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | $[2]$ | |
| Roots of unity: | $240 = (3^{ 4 } - 1) \cdot 3$ |
|
Intermediate fields
| $\Q_{3}(\sqrt{2})$, $\Q_{3}(\sqrt{3})$, $\Q_{3}(\sqrt{3\cdot 2})$, 3.4.1.0a1.1, 3.2.2.2a1.2, 3.2.2.2a1.1, 3.1.4.3a1.1 x2, 3.1.4.3a1.2 x2, 3.4.2.4a1.2, 3.2.4.6a1.3, 3.2.4.6a1.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 3.4.1.0a1.1 $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{4} + 2 x^{3} + 2 \)
|
|
| Relative Eisenstein polynomial: |
\( x^{4} + 3 \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^3 + z^2 + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $16$ |
| Galois group: | $C_4:C_4$ (as 16T8) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.75$ |
| Galois splitting model: | not computed |