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I provide a systematic construction of points, defined over finite radicalextensions, on any Legendre curve over any field of characteristic not equaltwo. This includes as special case Douglas Ulmer's construction of rationalpoints over a rational function field in characteristic $p>0$. In particular Ishow that if $n\geq 4$ is any even integer and not divisible by thecharacteristic of the field then any elliptic curve $E$ over this field has atleast $2n$ rational points over a finite solvable field extension. Underadditional hypothesis, when the ground field is a number field, I show thatthese are of infinite order. I also show that Ulmer's points lift tocharacteristic zero and in particular to the canonical lifting.

A method for construction of rational points over elliptic curves II: Points over solvable extensions