Base \(\Q_{3}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(7\)
Galois group $QD_{16}$ (as 8T8)

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Defining polynomial

\(x^{8} + 3\) Copy content Toggle raw display


Base field: $\Q_{3}$
Degree $d$: $8$
Ramification exponent $e$: $8$
Residue field degree $f$: $1$
Discriminant exponent $c$: $7$
Discriminant root field: $\Q_{3}(\sqrt{3})$
Root number: $i$
$\card{ \Aut(K/\Q_{ 3 }) }$: $2$
This field is not Galois over $\Q_{3}.$
Visible slopes:None

Intermediate fields

$\Q_{3}(\sqrt{3\cdot 2})$,

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{3}$
Relative Eisenstein polynomial: \( x^{8} + 3 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{7} + 2z^{6} + z^{5} + 2z^{4} + z^{3} + 2z^{2} + z + 2$
Associated inertia:$2$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$\SD_{16}$ (as 8T8)
Inertia group:$C_8$ (as 8T1)
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$8$
Wild slopes:None
Galois mean slope:$7/8$
Galois splitting model:$x^{8} + 24 x^{6} + 210 x^{4} + 792 x^{2} + 1083$