Defining polynomial
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$( x^{2} + 4 x + 2 )^{4} + 10 ( x^{2} + 4 x + 2 )^{2} + 160 ( x^{2} + 4 x + 2 ) + 345$
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Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $6$ |
| Discriminant root field: | $\Q_{5}$ |
| Root number: | $-1$ |
| $\card{ \Gal(K/\Q_{ 5 }) }$: | $8$ |
| This field is Galois and abelian over $\Q_{5}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{5}(\sqrt{2})$, $\Q_{5}(\sqrt{5})$, $\Q_{5}(\sqrt{5\cdot 2})$, 5.4.2.1, 5.4.3.2, 5.4.3.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{5}(\sqrt{2})$ $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{2} + 4 x + 2 \)
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| Relative Eisenstein polynomial: |
\( x^{4} + 5 \)
$\ \in\Q_{5}(t)[x]$
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Ramification polygon
Not computedInvariants of the Galois closure
| Galois group: | $C_2\times C_4$ (as 8T2) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $4$ |
| Wild slopes: | None |
| Galois mean slope: | $3/4$ |
| Galois splitting model: | $x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1$ |