Base \(\Q_{23}\)
Degree \(10\)
e \(5\)
f \(2\)
c \(8\)
Galois group $F_5$ (as 10T4)

Related objects

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Defining polynomial

\( x^{10} - 23 x^{5} + 3703 \)


Base field: $\Q_{23}$
Degree $d$ : $10$
Ramification exponent $e$ : $5$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $8$
Discriminant root field: $\Q_{23}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 23 })|$: $2$
This field is not Galois over $\Q_{23}$.

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:$\Q_{23}(\sqrt{*})$ $\cong \Q_{23}(t)$ where $t$ is a root of \( x^{2} - x + 7 \)
Relative Eisenstein polynomial:$ x^{5} - 23 t \in\Q_{23}(t)[x]$

Invariants of the Galois closure

Galois group:$F_5$ (as 10T4)
Inertia group:Intransitive group isomorphic to $C_5$
Unramified degree:$4$
Tame degree:$5$
Wild slopes:None
Galois mean slope:$4/5$
Galois splitting model:Not computed