Base \(\Q_{2}\)
Degree \(15\)
e \(5\)
f \(3\)
c \(12\)
Galois group $F_5\times C_3$ (as 15T8)

Related objects

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Defining polynomial

\( x^{15} - 4 x^{5} + 8 \)


Base field: $\Q_{2}$
Degree $d$: $15$
Ramification exponent $e$: $5$
Residue field degree $f$: $3$
Discriminant exponent $c$: $12$
Discriminant root field: $\Q_{2}(\sqrt{5})$
Root number: $1$
$|\Aut(K/\Q_{ 2 })|$: $3$
This field is not Galois over $\Q_{2}.$

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: $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{3} - x + 1 \)
Relative Eisenstein polynomial:$ x^{5} - 2 t \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_3\times F_5$ (as 15T8)
Inertia group:Intransitive group isomorphic to $C_5$
Unramified degree:$12$
Tame degree:$5$
Wild slopes:None
Galois mean slope:$4/5$
Galois splitting model:Not computed