Defining polynomial
\(x^{4} + 3 x^{2} + 12 x + 2\)
|
Invariants
Base field: | $\Q_{13}$ |
Degree $d$: | $4$ |
Ramification exponent $e$: | $1$ |
Residue field degree $f$: | $4$ |
Discriminant exponent $c$: | $0$ |
Discriminant root field: | $\Q_{13}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 13 }) }$: | $4$ |
This field is Galois and abelian over $\Q_{13}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{13}(\sqrt{2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 13.4.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of
\( x^{4} + 3 x^{2} + 12 x + 2 \)
|
Relative Eisenstein polynomial: |
\( x - 13 \)
$\ \in\Q_{13}(t)[x]$
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Ramification polygon
The ramification polygon is trivial for unramified extensions.