Properties

Label 19.10.0.1
Base \(\Q_{19}\)
Degree \(10\)
e \(1\)
f \(10\)
c \(0\)
Galois group $C_{10}$ (as 10T1)

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Defining polynomial

\(x^{10} + 18 x^{5} + 13 x^{4} + 17 x^{3} + 3 x^{2} + 4 x + 2\) Copy content Toggle raw display

Invariants

Base field: $\Q_{19}$
Degree $d$: $10$
Ramification exponent $e$: $1$
Residue field degree $f$: $10$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{19}(\sqrt{2})$
Root number: $1$
$\card{ \Gal(K/\Q_{ 19 }) }$: $10$
This field is Galois and abelian over $\Q_{19}.$
Visible slopes:None

Intermediate fields

$\Q_{19}(\sqrt{2})$, 19.5.0.1

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

Unramified/totally ramified tower

Unramified subfield:19.10.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{10} + 18 x^{5} + 13 x^{4} + 17 x^{3} + 3 x^{2} + 4 x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x - 19 \) $\ \in\Q_{19}(t)[x]$ Copy content Toggle raw display

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

Galois group:$C_{10}$ (as 10T1)
Inertia group:trivial
Wild inertia group:$C_1$
Unramified degree:$10$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$