Base \(\Q_{11}\)
Degree \(15\)
e \(1\)
f \(15\)
c \(0\)
Galois group $C_{15}$ (as 15T1)

Related objects


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Defining polynomial

\(x^{15} + 10 x^{6} + 7 x^{5} + 5 x^{3} + 9\) Copy content Toggle raw display


Base field: $\Q_{11}$
Degree $d$: $15$
Ramification exponent $e$: $1$
Residue field degree $f$: $15$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{11}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 11 }) }$: $15$
This field is Galois and abelian over $\Q_{11}.$
Visible slopes:None

Intermediate fields,

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

Unramified/totally ramified tower

Unramified subfield: $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{15} + 10 x^{6} + 7 x^{5} + 5 x^{3} + 9 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x - 11 \) $\ \in\Q_{11}(t)[x]$ Copy content Toggle raw display

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

Galois group:$C_{15}$ (as 15T1)
Inertia group:trivial
Wild inertia group:$C_1$
Unramified degree:$15$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:Not computed