Defining polynomial
\(x^{2} - x + 6\)  |
Invariants
| Base field: | $\Q_{157}$ |
| Degree $d$: | $2$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{157}(\sqrt{2})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 157 })|$: | $2$ |
| This field is Galois and abelian over $\Q_{157}.$ | |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 157 }$. |
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{157}(\sqrt{2})$ $\cong \Q_{157}(t)$ where $t$ is a root of \( x^{2} - x + 6 \) |
| Relative Eisenstein polynomial: | \( x - 157 \)$\ \in\Q_{157}(t)[x]$ |
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: | $x^{2} - x + 6$ |