Defining polynomial
|
\(x^{5} + 2 x + 1\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $5$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 3 }) }$: | $5$ |
| 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: | 3.5.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{5} + 2 x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x - 3 \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.