Number Theory
In the number theory of the twentieth century, the infinite descent method was taken up again, and pushed to a point where it connected with the main thrust of algebraic number theory and the study of L-functions. The structural result of Mordell, that the rational points on an elliptic curve E form a finitely-generated abelian group, used an infinite descent argument based on E/2E in Fermat's style.
To extend this to the case of an abelian variety A, André Weil had to make more explicit the way of quantifying the size of a solution, by means of a height function – a concept that became foundational. To show that A(Q)/2A(Q) is finite, which is certainly a necessary condition for the finite generation of the group A(Q) of rational points of A, one must do calculations in what later was recognised as Galois cohomology. In this way, abstractly-defined cohomology groups in the theory become identified with descents in the tradition of Fermat. The Mordell–Weil theorem was at the start of what later became a very extensive theory.
Read more about this topic: Proof By Infinite Descent
Famous quotes containing the words number and/or theory:
“At thirty years a woman asks her lover to give her back the esteem she has forfeited for his sake; she lives only for him, her thoughts are full of his future, he must have a great career, she bids him make it glorious; she can obey, entreat, command, humble herself, or rise in pride; times without number she brings comfort when a young girl can only make moan.”
—Honoré De Balzac (1799–1850)
“A theory if you hold it hard enough
And long enough gets rated as a creed....”
—Robert Frost (1874–1963)