2017-08-29 23-53.pdf - Hum aef fixma= 1£ a= W ole(1 var:0...

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Unformatted text preview: Hum aef fixma=+ . 1£+a=-_ - W. ole? (1+ var-:0 0F. .\As. " Le’raEFStM a= SVFXE'I. hero «with ’ O‘FX .ISH‘O'ZQ A ppendix C Fields 48? Definitions. A field F is a set on which two operations + and - (called addition and multiplication, respectively) are defined so that, far each pair of elements 11:, y in F, there are unique elements :1: +31 and x-y in F for Which the following conditions hold for: all elements a, b, c in F, (F 1) (1+5: b+a and a-b=b‘a. (commutativity of addition and multiplication) (Fn(a+w+c=a+lb+d zmd mtyc=a4ad (associativity of addition and multiplication) (F 3) There eXZiSt distinct elements 0 and 1 in F such that 0+a=a and 1-a=a (existence of identity elements for addition and multiplication) (F 4) Ear each element a in F and each nonzero element b in F, there exist elements c and d in F such that a+c=0 and b~d=1 (existence of inverses for addition and multiplication) (F 5) a-(b + c) = a-b + a-c (distributivity of multiplication over addition) The elements at + y and wry are called the sum and product, respectively, ofa; and y. The elements 0 (read “zero” ) and 1 ”(read “one”) mentioned in (F 3) are called identity elements for addition and multiplication, reSpec- tively, and the elements c and d referred to in (F 4) are called an additive inverse for a and a multiplicative inverse for b, respectively. Example 1 p p The set of real numbers R ‘With the usual definitions of addition and multi- plication is a field. 9 Example 2 The set (if rational numbers with the usual definitions of plication is a field. § addition and multi- Example 3 The set of all real numbers of the form a + bx/i, .‘where a. and b are rational numbers, with addition and multiplication as in ‘R is a field. 9 Example 4 The field Z2 consists of two elements 0 and 1 wit and multiplication defined by the equations O+0=m 0+1=1+0=L WO=Q 01=L0=m mm L1=L h the operations of addition 1+1=05 O ...
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