Defining polynomial
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\(x^{7} + 2 x^{2} + 1\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $7$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $7$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 3 }) }$: | $7$ |
| This field is Galois and abelian over $\Q_{3}.$ | |
| Visible slopes: | None |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 3 }$. |
Unramified/totally ramified tower
| Unramified subfield: | 3.7.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{7} + 2 x^{2} + 1 \)
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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: | $C_7$ (as 7T1) |
| Inertia group: | trivial |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $7$ |
| Tame degree: | $1$ |
| Wild slopes: | None |
| Galois mean slope: | $0$ |
| Galois splitting model: |
$x^{7} + x^{6} - 12 x^{5} - 7 x^{4} + 28 x^{3} + 14 x^{2} - 9 x + 1$
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