Defining polynomial
\(x^{2} - x + 3\) ![]() |
Invariants
Base field: | $\Q_{43}$ |
Degree $d$: | $2$ |
Ramification exponent $e$: | $1$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $0$ |
Discriminant root field: | $\Q_{43}(\sqrt{2})$ |
Root number: | $1$ |
$|\Gal(K/\Q_{ 43 })|$: | $2$ |
This field is Galois and abelian over $\Q_{43}.$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 43 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{43}(\sqrt{2})$ $\cong \Q_{43}(t)$ where $t$ is a root of \( x^{2} - x + 3 \) ![]() |
Relative Eisenstein polynomial: | \( x - 43 \)$\ \in\Q_{43}(t)[x]$ ![]() |
Invariants of the Galois closure
Galois group: | $C_2$ (as 2T1) |
Inertia group: | trivial |
Unramified degree: | $2$ |
Tame degree: | $1$ |
Wild slopes: | None |
Galois mean slope: | $0$ |
Galois splitting model: | $x^{2} - x + 3$ |