Algebraic Number Theory

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)

    This nightmare occupied some ten pages of manuscript and wound off with a sermon so destructive of all hope to non-Presbyterians that it took the first prize. This composition was considered to be the very finest effort of the evening.... It may be remarked, in passing, that the number of compositions in which the word “beauteous” was over-fondled, and human experience referred to as “life’s page,” was up to the usual average.
    Mark Twain [Samuel Langhorne Clemens] (1835–1910)

    There is in him, hidden deep-down, a great instinctive artist, and hence the makings of an aristocrat. In his muddled way, held back by the manacles of his race and time, and his steps made uncertain by a guiding theory which too often eludes his own comprehension, he yet manages to produce works of unquestionable beauty and authority, and to interpret life in a manner that is poignant and illuminating.
    —H.L. (Henry Lewis)