Defining polynomial
\( x^{2} - x + 12 \) |
Invariants
Base field: | $\Q_{67}$ |
Degree $d$: | $2$ |
Ramification exponent $e$: | $1$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $0$ |
Discriminant root field: | $\Q_{67}(\sqrt{2})$ |
Root number: | $1$ |
$|\Gal(K/\Q_{ 67 })|$: | $2$ |
This field is Galois and abelian over $\Q_{67}.$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 67 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{67}(\sqrt{2})$ $\cong \Q_{67}(t)$ where $t$ is a root of \( x^{2} - x + 12 \) |
Relative Eisenstein polynomial: | $ x - 67 \in\Q_{67}(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: | $x^{2} - x + 12$ |