Defining polynomial
\(x + 27\) |
Invariants
Base field: | $\Q_{29}$ |
Degree $d$: | $1$ |
Ramification exponent $e$: | $1$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $0$ |
Discriminant root field: | $\Q_{29}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 29 }) }$: | $1$ |
This field is Galois and abelian over $\Q_{29}.$ | |
Visible slopes: | None |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 29 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{29}$ |
Relative Eisenstein polynomial: | \( x - 29 \) |
Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
Galois group: | $C_1$ (as 1T1) |
Inertia group: | $C_1$ (as 1T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $1$ |
Tame degree: | $1$ |
Wild slopes: | None |
Galois mean slope: | $0$ |
Galois splitting model: | $x + 2$ |