Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization, the behaviour of ideals, and field extensions. In this setting, the familiar features of the integers—such as unique factorization—need not hold. The virtue of the primary machinery employed—Galois theory, group cohomology, group representations, and L-functions—is that it allows one to deal with new phenomena and yet partially recover the behaviour of the usual integers.
Famous quotes containing the words algebraic, number and/or theory:
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)
“Strange goings on! Jones did it slowly, deliberately, in the bathroom, with a knife, at midnight. What he did was butter a piece of toast. We are too familiar with the language of action to notice at first an anomaly: the ‘it’ of ‘Jones did it slowly, deliberately,...’ seems to refer to some entity, presumably an action, that is then characterized in a number of ways.”
—Donald Davidson (b. 1917)
“It is not enough for theory to describe and analyse, it must itself be an event in the universe it describes. In order to do this theory must partake of and become the acceleration of this logic. It must tear itself from all referents and take pride only in the future. Theory must operate on time at the cost of a deliberate distortion of present reality.”
—Jean Baudrillard (b. 1929)