Defining polynomial
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\(x^{10} + 2 x^{6} + 2 x^{5} + 2 x^{4} + x + 2\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $10$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 3 }) }$: | $10$ |
| This field is Galois and abelian over $\Q_{3}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 3.10.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{10} + 2 x^{6} + 2 x^{5} + 2 x^{4} + x + 2 \)
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| Relative Eisenstein polynomial: |
\( x - 3 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
| Galois group: | $C_{10}$ (as 10T1) |
| Inertia group: | trivial |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $10$ |
| Tame degree: | $1$ |
| Wild slopes: | None |
| Galois mean slope: | $0$ |
| Galois splitting model: |
$x^{10} + x^{9} + 2 x^{8} - 16 x^{7} - 9 x^{6} - 11 x^{5} + 43 x^{4} + 6 x^{3} + 63 x^{2} + 20 x + 25$
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