Defining polynomial
\(x^{6} + 5\) |
Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $5$ |
Discriminant root field: | $\Q_{5}(\sqrt{5})$ |
Root number: | $-1$ |
$\card{ \Aut(K/\Q_{ 5 }) }$: | $2$ |
This field is not Galois over $\Q_{5}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{5}(\sqrt{5})$, 5.3.2.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{5}$ |
Relative Eisenstein polynomial: | \( x^{6} + 5 \) |
Ramification polygon
Data not computedInvariants of the Galois closure
Galois group: | $D_6$ (as 6T3) |
Inertia group: | $C_6$ (as 6T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $6$ |
Wild slopes: | None |
Galois mean slope: | $5/6$ |
Galois splitting model: | $x^{6} - 5$ |