Base \(\Q_{11}\)
Degree \(9\)
e \(9\)
f \(1\)
c \(8\)
Galois group $(C_9:C_3):C_2$ (as 9T10)

Related objects


Learn more

Defining polynomial

\(x^{9} + 11\) Copy content Toggle raw display


Base field: $\Q_{11}$
Degree $d$: $9$
Ramification exponent $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
Discriminant root field: $\Q_{11}$
Root number: $1$
$\card{ \Aut(K/\Q_{ 11 }) }$: $1$
This field is not Galois 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:$\Q_{11}$
Relative Eisenstein polynomial: \( x^{9} + 11 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{8} + 9z^{7} + 3z^{6} + 7z^{5} + 5z^{4} + 5z^{3} + 7z^{2} + 3z + 9$
Associated inertia:$6$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_9:C_6$ (as 9T10)
Inertia group:$C_9$ (as 9T1)
Wild inertia group:$C_1$
Unramified degree:$6$
Tame degree:$9$
Wild slopes:None
Galois mean slope:$8/9$
Galois splitting model:$x^{9} - 11$