SELECTED CHAPTERS FROM ALGEBRA I. R. Shafarevich Preface

Selected chapters from algebra 3(Remark. There exists exactly one such number. If 0 0 were another numberwith the same property, wewould have 0 0 +0=0 0 ,by the denition of 0, 0 0 +0=0+0 0 by the commutative lawand0+0 0 =0,by the denition of 0 0 . Finally, weobtain 0 0 =0 0 +0=0+0 0 = 0, i.e., 0 0 =0.)I 4 . For each real number a there exists a number called opposite, denoted by ;a,such that a +(;a) =0.(Remark. For the given number a there exists exactly one such number.If a 0 were another number with the same property: a + a 0 = 0, we would have(a +(;a)) + a 0 =0+a 0 = a 0 . Also, (a +(;a)) + a 0 =((;a)+a)+a 0 ,andbytheassociative law, ((;a)+a)+a 0 =(;a)+(a + a 0 ). By the property ofnumber a 0 ,a + a 0 =0and(;a) +0=;a. Taking these equalities together, we obtain thata 0 = ;a.)II (axioms of multiplication)II 1 . Commutative law: ab = ba for arbitrary real numbers a and b.II 2 . Associative law: a(bc) =(ab)c for arbitrary real numbers a, b and c.II 3 . There exists a number called unit, denoted by 1,such that a 1=a for anarbitrary real number a.(Remark. There exists only one such number. It can be proved in thesame way as the remark following axiom I 3 |we onlyhave to replace addition bymultiplication, and 0 by 1.)II 4 . For each realnumber a, dierent from 0, there exists a number called inverse,denoted by a ;1 ,suchthata a ;1 =1.(Remark. For each realnumber a dierent from 0, there exists only one suchnumber. The proof is exactly the same as in the remark following axiom I 4 .)III (axiom of addition and multiplication)III 1 . Distributive law: (a + b)c = ac + bc for arbitrary real numbers a, b and c.IV (axioms of order)IV 1 . For any two real numbers a and b exactly one of the following three relationsholds: a = b or a