Base \(\Q_{3}\)
Degree \(10\)
e \(10\)
f \(1\)
c \(9\)
Galois group $F_{5}\times C_2$ (as 10T5)

Related objects


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Defining polynomial

\(x^{10} + 6\) Copy content Toggle raw display


Base field: $\Q_{3}$
Degree $d$: $10$
Ramification exponent $e$: $10$
Residue field degree $f$: $1$
Discriminant exponent $c$: $9$
Discriminant root field: $\Q_{3}(\sqrt{3})$
Root number: $i$
$\card{ \Aut(K/\Q_{ 3 }) }$: $2$
This field is not Galois over $\Q_{3}.$
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:$\Q_{3}$
Relative Eisenstein polynomial: \( x^{10} + 6 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{9} + z^{8} + 1$
Associated inertia:$4$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_2\times F_5$ (as 10T5)
Inertia group:$C_{10}$ (as 10T1)
Wild inertia group:$C_1$
Unramified degree:$4$
Tame degree:$10$
Wild slopes:None
Galois mean slope:$9/10$
Galois splitting model:$x^{10} - 3$