Abstract
Let k be a number field which contains the fifth roots of unity and K/k an A5-extension. According to Klein and Serre K/k can be described by adjunction of the 5-torsion points of a suitable elliptic curve E. It is well known that there exists such a curve E defined over k if and only if there exists a quadratic overfield M of K such that M is Galois over k with group <5. This corresponds to a 2-dimensional Galois representation of the Galois group G(k) of k with trivial determinant.
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