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