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: | $x^{3} + 3 x + 3$ |