Defining polynomial
\(x^{4} + 3\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $4$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $3$ |
Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
Root number: | $-i$ |
$\card{ \Aut(K/\Q_{ 3 }) }$: | $2$ |
This field is not Galois over $\Q_{3}.$ | |
Visible slopes: | None |
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.
Unramified/totally ramified 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 group: | $D_4$ (as 4T3) |
Inertia group: | $C_4$ (as 4T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $4$ |
Wild slopes: | None |
Galois mean slope: | $3/4$ |
Galois splitting model: | $x^{4} + 3$ |