Defining polynomial
|
\(x^{2} + 74\)
|
Invariants
| Base field: | $\Q_{37}$ |
| Degree $d$: | $2$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $1$ |
| Discriminant root field: | $\Q_{37}(\sqrt{37\cdot 2})$ |
| Root number: | $-1$ |
| $\card{ \Gal(K/\Q_{ 37 }) }$: | $2$ |
| This field is Galois and abelian over $\Q_{37}.$ | |
| Visible slopes: | None |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 37 }$. |
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{37}$ |
| Relative Eisenstein polynomial: |
\( x^{2} + 74 \)
|
Ramification polygon
| Residual polynomials: | $z + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_2$ (as 2T1) |
| Inertia group: | $C_2$ (as 2T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $1$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | $x^{2} + 74$ |