Defining polynomial
\(x^{2} + 2 x + 2\) ![]() |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $2$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $2$ |
Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
Root number: | $i$ |
$|\Gal(K/\Q_{ 2 })|$: | $2$ |
This field is Galois and abelian over $\Q_{2}.$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 2 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{2}$ |
Relative Eisenstein polynomial: | \( x^{2} + 2 x + 2 \) ![]() |
Invariants of the Galois closure
Galois group: | $C_2$ (as 2T1) |
Inertia group: | $C_2$ |
Unramified degree: | $1$ |
Tame degree: | $1$ |
Wild slopes: | [2] |
Galois mean slope: | $1$ |
Galois splitting model: | $x^{2} + 2 x + 2$ ![]() |