Defining polynomial
\(x^{3} + 3 x + 3\) ![]() |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $3$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $3$ |
Discriminant root field: | $\Q_{3}(\sqrt{3\cdot 2})$ |
Root number: | $i$ |
$|\Aut(K/\Q_{ 3 })|$: | $1$ |
This field is not Galois over $\Q_{3}.$ |
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: | \( x^{3} + 3 x + 3 \) ![]() |
Invariants of the Galois closure
Galois group: | $S_3$ (as 3T2) |
Inertia group: | $S_3$ |
Unramified degree: | $1$ |
Tame degree: | $2$ |
Wild slopes: | [3/2] |
Galois mean slope: | $7/6$ |
Galois splitting model: | $x^{3} + 3 x + 3$ ![]() |