Defining polynomial
|
\(x^{4} + 3\)
|
Invariants
| Base field: | $\Q_{3}$ |
|
| Degree $d$: | $4$ |
|
| Ramification index $e$: | $4$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $3$ |
|
| Discriminant root field: | $\Q_{3}(\sqrt{3})$ | |
| Root number: | $-i$ | |
| $\Aut(K/\Q_{3})$: | $C_2$ | |
| This field is not Galois over $\Q_{3}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | $[2]$ | |
| Roots of unity: | $6 = (3 - 1) \cdot 3$ |
|
Intermediate fields
| $\Q_{3}(\sqrt{3\cdot 2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{3}$ |
|
| Relative Eisenstein polynomial: |
\( x^{4} + 3 \)
|
Ramification polygon
| Residual polynomials: | $z^3 + z^2 + 1$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $8$ |
| Galois group: | $D_4$ (as 4T3) |
| Inertia group: | $C_4$ (as 4T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.75$ |
| Galois splitting model: | $x^{4} - 3 x^{2} + 3$ |