Properties

Label 2.10.14.5
Base \(\Q_{2}\)
Degree \(10\)
e \(10\)
f \(1\)
c \(14\)
Galois group $F_{5}\times C_2$ (as 10T5)

Related objects

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Defining polynomial

\(x^{10} - 2 x^{5} - 2\)  Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $10$
Ramification exponent $e$: $10$
Residue field degree $f$: $1$
Discriminant exponent $c$: $14$
Discriminant root field: $\Q_{2}(\sqrt{-5})$
Root number: $i$
$|\Aut(K/\Q_{ 2 })|$: $2$
This field is not Galois over $\Q_{2}.$

Intermediate fields

$\Q_{2}(\sqrt{-5})$, 2.5.4.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}$
Relative Eisenstein polynomial:\( x^{10} - 160 x^{9} - 40460 x^{8} - 2808640 x^{7} - 98032360 x^{6} - 998918222 x^{5} - 8072348440 x^{4} + 192345129720 x^{3} - 1170688441400 x^{2} + 3109663379160 x - 3510500485802 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_2\times F_5$ (as 10T5)
Inertia group:$C_{10}$
Unramified degree:$4$
Tame degree:$5$
Wild slopes:[2]
Galois mean slope:$7/5$
Galois splitting model:$x^{10} + 5 x^{8} + 10 x^{6} + 10 x^{4} + 5 x^{2} + 5$