Defining polynomial
\(x + 4\) ![]() |
Invariants
Base field: | $\Q_{67}$ |
Degree $d$: | $1$ |
Ramification exponent $e$: | $1$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $0$ |
Discriminant root field: | $\Q_{67}$ |
Root number: | $1$ |
$|\Gal(K/\Q_{ 67 })|$: | $1$ |
This field is Galois and abelian over $\Q_{67}.$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 67 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{67}$ |
Relative Eisenstein polynomial: | \( x - 67 \) ![]() |
Invariants of the Galois closure
Galois group: | $C_1$ (as 1T1) |
Inertia group: | Trivial |
Unramified degree: | $1$ |
Tame degree: | $1$ |
Wild slopes: | None |
Galois mean slope: | $0$ |
Galois splitting model: | $x + 4$ ![]() |