Defining polynomial
\(x^{2} + 4 x + 2\) |
Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $2$ |
Ramification exponent $e$: | $1$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $0$ |
Discriminant root field: | $\Q_{5}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 5 }) }$: | $2$ |
This field is Galois and abelian over $\Q_{5}.$ | |
Visible slopes: | None |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 5 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{5}(\sqrt{2})$ $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{2} + 4 x + 2 \) |
Relative Eisenstein polynomial: | \( x - 5 \) $\ \in\Q_{5}(t)[x]$ |
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: | $x^{2} - x + 2$ |