Defining polynomial
|
\(x^{2} + x + 1\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $2$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 2 }) }$: | $2$ |
| This field is Galois and abelian over $\Q_{2}.$ | |
| Visible slopes: | None |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 2 }$. |
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{2} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x - 2 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
| Galois group: | $C_2$ (as 2T1) |
| Inertia group: | trivial |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $1$ |
| Wild slopes: | None |
| Galois mean slope: | $0$ |
| Galois splitting model: | $x^{2} - x + 1$ |
Additional information
The image of the $2$-adic logarithm on $1$-units is $$ \log_2(1+M) = \{ 2\alpha + 4 \beta t \mid \alpha,\beta\in\Z_2\}$$ where $M$ is the maximal ideal in the ring of integers of this field and $t$ is a root of $x^2+x+1$.