Role of Field Theory In Pure Mathematics

The role of field theory in number theory and algebraic geometry

Abstract

Any field may be used as the scalars for a vectorspace, which is the standard general context for linearalgebra. The theory of fieldextensions (including Galois theory) involves the roots of polynomials with coefficients ina field; among other results, this theory leads to impossibility proofs for theclassical problems of angletrisection and squaring the circle with a compass and straightedge, as well as aproof of the Abel–Ruffini theorem on the algebraicinsolubility of quintic equations. In modern mathematics, thetheory of fields (or field theory) plays an essentialrole in number theory and algebraic geometry.