Properties

Label 17.11.0.1
Base \(\Q_{17}\)
Degree \(11\)
e \(1\)
f \(11\)
c \(0\)
Galois group $C_{11}$ (as 11T1)

Related objects

Downloads

Learn more

Defining polynomial

\(x^{11} + 5 x + 14\) Copy content Toggle raw display

Invariants

Base field: $\Q_{17}$
Degree $d$: $11$
Ramification exponent $e$: $1$
Residue field degree $f$: $11$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{17}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 17 }) }$: $11$
This field is Galois and abelian over $\Q_{17}.$
Visible slopes:None

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 17 }$.

Unramified/totally ramified tower

Unramified subfield:17.11.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of \( x^{11} + 5 x + 14 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x - 17 \) $\ \in\Q_{17}(t)[x]$ Copy content Toggle raw display

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

Galois group:$C_{11}$ (as 11T1)
Inertia group:trivial
Wild inertia group:$C_1$
Unramified degree:$11$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{11} - x^{10} - 10 x^{9} + 9 x^{8} + 36 x^{7} - 28 x^{6} - 56 x^{5} + 35 x^{4} + 35 x^{3} - 15 x^{2} - 6 x + 1$