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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 xy in F for Which the following conditions hold for: all elements a, b, c in F, (F 1) (1+5: b+a and ab=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 1a=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) = ab + ac
(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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 Fall '08
 Staff
 Linear Algebra, Algebra, Addition, Multiplication, HW 1

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