The Mordell–Weil Theorem states that the group of rational points on an abelian variety over a number field is finitely generated. This is significant in understanding the structure of rational solutions to algebraic equations, particularly in relation to elliptic curves and quadratic Diophantine equations. The theorem highlights how the rational solutions can be broken down into a finite number of generators and a torsion subgroup, revealing deep connections between number theory and algebraic geometry.
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