Defining polynomial
\( x^{2} + 3 \) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$ : | $2$ |
Ramification exponent $e$ : | $2$ |
Residue field degree $f$ : | $1$ |
Discriminant exponent $c$ : | $1$ |
Discriminant root field: | $\Q_{3}(\sqrt{3*})$ |
Root number: | $i$ |
$|\Gal(K/\Q_{ 3 })|$: | $2$ |
This field is Galois and abelian 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: | \( x^{2} + 3 \) |
Invariants of the Galois closure
Galois group: | $C_2$ (as 2T1) |
Inertia group: | $C_2$ |
Unramified degree: | $1$ |
Tame degree: | $2$ |
Wild slopes: | None |
Galois mean slope: | $1/2$ |
Galois splitting model: | $x^{2} + 3$ |