Properties

Label 3.10.8.1
Base \(\Q_{3}\)
Degree \(10\)
e \(5\)
f \(2\)
c \(8\)
Galois group $F_5$ (as 10T4)

Related objects

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Defining polynomial

\( x^{10} - 3 x^{5} + 18 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $10$
Ramification exponent $e$ : $5$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $8$
Discriminant root field: $\Q_{3}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 3 })|$: $2$
This field is not Galois over $\Q_{3}$.

Intermediate fields

$\Q_{3}(\sqrt{*})$, 3.5.4.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{3}(\sqrt{*})$ $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{2} - x + 2 \)
Relative Eisenstein polynomial:$ x^{5} - 3 t \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:$F_5$ (as 10T4)
Inertia group:Intransitive group isomorphic to $C_5$
Unramified degree:$4$
Tame degree:$5$
Wild slopes:None
Galois mean slope:$4/5$
Galois splitting model:$x^{10} - 5 x^{9} + 5 x^{8} + 10 x^{7} - 15 x^{6} - 8 x^{5} + 40 x^{3} - 35 x^{2} + 10 x - 4$