Galois theory and rational points on elliptic curves

What you've written down is relevant for finding rational torsion points on an elliptic curve. If that's what you want to do, Galois theory is certainly relevant. For instance, suppose you have an elliptic curve in Weierstrass form,

y^2 = f(x)

with f a cubic. Now suppose you find that f(x) has a linear factor (x-a). (I certainly take this to be a "Galois-theoretic" condition on f.) Then you've found a rational point of your curve, namely (a,0).

The relationship between Galois theory and points of infinite order is more subtle, involving Galois cohomology, and is discussed in chapter 10 of Silverman's book The Arithmetic of Elliptic Curves.