Defining polynomial
\(x^{2} + 26\) |
Invariants
Base field: | $\Q_{13}$ |
Degree $d$: | $2$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $1$ |
Discriminant root field: | $\Q_{13}(\sqrt{13\cdot 2})$ |
Root number: | $-1$ |
$\card{ \Gal(K/\Q_{ 13 }) }$: | $2$ |
This field is Galois and abelian over $\Q_{13}.$ | |
Visible slopes: | None |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 13 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{13}$ |
Relative Eisenstein polynomial: | \( x^{2} + 26 \) |
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: | $x^{2} + 26$ |