Defining polynomial
|
\(x^{15} + 2 x^{5} + 3 x^{3} + 3 x^{2} + 4 x + 3\)
|
Invariants
| Base field: | $\Q_{5}$ |
|
| Degree $d$: | $15$ |
|
| Ramification index $e$: | $1$ |
|
| Residue field degree $f$: | $15$ |
|
| Discriminant exponent $c$: | $0$ |
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| Discriminant root field: | $\Q_{5}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{5})$ $=$ $\Gal(K/\Q_{5})$: | $C_{15}$ | |
| This field is Galois and abelian over $\Q_{5}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $30517578124 = (5^{ 15 } - 1)$ |
|
Intermediate fields
| 5.3.1.0a1.1, 5.5.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 5.15.1.0a1.1 $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{15} + 2 x^{5} + 3 x^{3} + 3 x^{2} + 4 x + 3 \)
|
|
| Relative Eisenstein polynomial: |
\( x - 5 \)
$\ \in\Q_{5}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.