Defining polynomial
\(x^{4} + 2 x^{2} + 15 x + 2\) |
Invariants
Base field: | $\Q_{29}$ |
Degree $d$: | $4$ |
Ramification exponent $e$: | $1$ |
Residue field degree $f$: | $4$ |
Discriminant exponent $c$: | $0$ |
Discriminant root field: | $\Q_{29}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 29 }) }$: | $4$ |
This field is Galois and abelian over $\Q_{29}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{29}(\sqrt{2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 29.4.0.1 $\cong \Q_{29}(t)$ where $t$ is a root of \( x^{4} + 2 x^{2} + 15 x + 2 \) |
Relative Eisenstein polynomial: | \( x - 29 \) $\ \in\Q_{29}(t)[x]$ |
Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
Galois group: | $C_4$ (as 4T1) |
Inertia group: | trivial |
Wild inertia group: | $C_1$ |
Unramified degree: | $4$ |
Tame degree: | $1$ |
Wild slopes: | None |
Galois mean slope: | $0$ |
Galois splitting model: | $x^{4} - x^{3} - 6 x^{2} + x + 1$ |