Properties

Label 3.10.5.1
Base \(\Q_{3}\)
Degree \(10\)
e \(2\)
f \(5\)
c \(5\)
Galois group $C_{10}$ (as 10T1)

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Defining polynomial

\( x^{10} - 18 x^{6} + 81 x^{2} - 243 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $10$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $5$
Discriminant exponent $c$ : $5$
Discriminant root field: $\Q_{3}(\sqrt{3})$
Root number: $-i$
$|\Gal(K/\Q_{ 3 })|$: $10$
This field is Galois and abelian over $\Q_{3}$.

Intermediate fields

$\Q_{3}(\sqrt{3})$, 3.5.0.1

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

Unramified/totally ramified tower

Unramified subfield:3.5.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{5} - x + 1 \)
Relative Eisenstein polynomial:$ x^{2} - 3 t^{2} \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{10}$ (as 10T1)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$5$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{10} - x^{9} - 200 x^{8} + 194 x^{7} + 11877 x^{6} - 3991 x^{5} - 248665 x^{4} - 32090 x^{3} + 1913681 x^{2} + 788162 x - 3646277$