SELECTED CHAPTERS FROM ALGEBRA I. R. Shafarevich Preface

2 I. R. Shafarevichsomewhere, and we knowwhich points lie on which lines or which triples of pointsA, B, C on the line l are such that B lies between A and C. And only in thirdplace there are axioms, i.e., statements about basic notions and relations amongthem. For instance: each two distinct points belong to exactly one line. Or: amongthree distinct points on a line, there is exactly one lying between other two.There is a complete analogy with real numbers. The basic notions here arereal numbers themselves. This means that, for the moment, we do not assumeanything more about real numbers, but only that they constitute a certain set.Basic relations between real numbers are of two dierent types: operations andinequalities. Let us describe them in more detail.1) Operations with real numbersFor every two realnumbers a and b we dene a third number c, called the sumof a and b. We write this as: a + b = c.For every two real numbers a and b we dene a third number d, called theproduct of a and b. We write this as: ab = d.2) Inequalities between real numbersFor some pairs of real numbers a and b we have thata is less than b. We writethis as: aa. If we want tosaythata a).Before we pass to the formulation of axioms connecting basic notions withbasic relations among them, let us emphasize once more the analogy with geometry.Write analogous notions in the table:AlgebraGeometryBasic notionsReal numbers Point, line, . . .Basic relationssum: a + b = cA point lies on a line.product: ab = dPoint C lies betweeninequality: a