Properties

Label 5.15.0.1
Base \(\Q_{5}\)
Degree \(15\)
e \(1\)
f \(15\)
c \(0\)
Galois group $C_{15}$ (as 15T1)

Related objects

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Defining polynomial

\( x^{15} + x^{2} + 2 \)

Invariants

Base field: $\Q_{5}$
Degree $d$: $15$
Ramification exponent $e$: $1$
Residue field degree $f$: $15$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{5}$
Root number: $1$
$|\Gal(K/\Q_{ 5 })|$: $15$
This field is Galois and abelian over $\Q_{5}.$

Intermediate fields

5.3.0.1, 5.5.0.1

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

Unramified/totally ramified tower

Unramified subfield:5.15.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{15} + x^{2} + 2 \)
Relative Eisenstein polynomial:$ x - 5 \in\Q_{5}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{15}$ (as 15T1)
Inertia group:trivial
Unramified degree:$15$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{15} - x^{14} - 22 x^{13} + 17 x^{12} + 166 x^{11} - 102 x^{10} - 533 x^{9} + 270 x^{8} + 729 x^{7} - 352 x^{6} - 393 x^{5} + 173 x^{4} + 80 x^{3} - 27 x^{2} - 6 x + 1$