Base \(\Q_{5}\)
Degree \(9\)
e \(9\)
f \(1\)
c \(8\)
Galois group $(C_9:C_3):C_2$ (as 9T10)

Related objects

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Defining polynomial

\(x^{9} - 5\)  Toggle raw display


Base field: $\Q_{5}$
Degree $d$: $9$
Ramification exponent $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
Discriminant root field: $\Q_{5}$
Root number: $1$
$|\Aut(K/\Q_{ 5 })|$: $1$
This field is not Galois over $\Q_{5}.$

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:$\Q_{5}$
Relative Eisenstein polynomial:\( x^{9} - 5 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$D_9:C_3$ (as 9T10)
Inertia group:$C_9$
Unramified degree:$6$
Tame degree:$9$
Wild slopes:None
Galois mean slope:$8/9$
Galois splitting model:$x^{9} - 5$