Defining polynomial
|
\(x^{4} + 5\)
|
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $4$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $3$ |
| Discriminant root field: | $\Q_{5}(\sqrt{5})$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{5})$ $=$$\Gal(K/\Q_{5})$: | $C_4$ |
| This field is Galois and abelian over $\Q_{5}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | $[1]$ |
| Roots of unity: | $20 = (5 - 1) \cdot 5$ |
Intermediate fields
| $\Q_{5}(\sqrt{5})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{5}$ |
| Relative Eisenstein polynomial: |
\( x^{4} + 5 \)
|
Ramification polygon
| Residual polynomials: | $z^{3} + 4 z^{2} + z + 4$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $4$ |
| Galois group: | $C_4$ (as 4T1) |
| Inertia group: | $C_4$ (as 4T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $1$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[]$ |
| Galois mean slope: | $0.75$ |
| Galois splitting model: | $x^{4} - x^{3} + x^{2} - x + 1$ |