Properties

Label 3.10.0.1
Base \(\Q_{3}\)
Degree \(10\)
e \(1\)
f \(10\)
c \(0\)
Galois group $C_{10}$ (as 10T1)

Related objects

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Defining polynomial

\(x^{10} - x^{3} - x + 2\)  Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $10$
Ramification exponent $e$: $1$
Residue field degree $f$: $10$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{3}(\sqrt{2})$
Root number: $1$
$|\Gal(K/\Q_{ 3 })|$: $10$
This field is Galois and abelian over $\Q_{3}.$

Intermediate fields

$\Q_{3}(\sqrt{2})$, 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:3.10.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{10} - x^{3} - x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x - 3 \)$\ \in\Q_{3}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_{10}$ (as 10T1)
Inertia group:trivial
Unramified degree:$10$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{10} + x^{9} + 2 x^{8} - 16 x^{7} - 9 x^{6} - 11 x^{5} + 43 x^{4} + 6 x^{3} + 63 x^{2} + 20 x + 25$  Toggle raw display