Defining polynomial
\(x^{4} + 2 x^{3} + 2\)
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Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $4$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[2, 2]$ |
Intermediate fields
$\Q_{2}(\sqrt{-1})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{2}$ |
Relative Eisenstein polynomial: |
\( x^{4} + 2 x^{3} + 2 \)
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Ramification polygon
Not computedInvariants of the Galois closure
Galois group: | $D_4$ (as 4T3) |
Inertia group: | $C_2^2$ (as 4T2) |
Wild inertia group: | $C_2^2$ |
Unramified degree: | $2$ |
Tame degree: | $1$ |
Wild slopes: | $[2, 2]$ |
Galois mean slope: | $3/2$ |
Galois splitting model: |
$x^{4} + 2 x^{3} + 2$
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