Base \(\Q_{23}\)
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

\(x^{10} + 115\) Copy content Toggle raw display


Base field: $\Q_{23}$
Degree $d$: $10$
Ramification exponent $e$: $10$
Residue field degree $f$: $1$
Discriminant exponent $c$: $9$
Discriminant root field: $\Q_{23}(\sqrt{23})$
Root number: $i$
$\card{ \Aut(K/\Q_{ 23 }) }$: $2$
This field is not Galois over $\Q_{23}.$
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:$\Q_{23}$
Relative Eisenstein polynomial: \( x^{10} + 115 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{9} + 10z^{8} + 22z^{7} + 5z^{6} + 3z^{5} + 22z^{4} + 3z^{3} + 5z^{2} + 22z + 10$
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: $x^{10} - 23$ Copy content Toggle raw display