Defining polynomial
\(x^{5} + 3\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $5$ |
Ramification exponent $e$: | $5$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $4$ |
Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
This field is not Galois 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: | $\Q_{3}$ |
Relative Eisenstein polynomial: | \( x^{5} + 3 \) |
Ramification polygon
Residual polynomials: | $z^{4} + 2z^{3} + z^{2} + z + 2$ |
Associated inertia: | $4$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $F_5$ (as 5T3) |
Inertia group: | $C_5$ (as 5T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $4$ |
Tame degree: | $5$ |
Wild slopes: | None |
Galois mean slope: | $4/5$ |
Galois splitting model: | $x^{5} - 3$ |