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BasicAlg

# BasicAlg - A-126 A.1 A.1.1 Basic Algebra Fields In doing...

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A-126 A.1 Basic Algebra A.1.1 Fields In doing coding theory it is advantageous for our alphabet to have a certain amount of mathematical structure. We are familiar at the bit level with boolean addition (EXCLUSIVE OR) and multiplication (AND) within the set { 0 , 1 } : + 0 1 0 0 1 1 1 0 × 0 1 0 0 0 1 0 1 We wish to give other alphabets, particularly ﬁnite ones, a workable arithmetic. The objects we study (and of which the set { 0 , 1 } together with the above operations is an example) are called ﬁelds. A ﬁeld is basically a set that possesses an arithmetic having (most of) the properties that we expect — the ability to add, multiply, subtract, and divide subject to the usual laws of commutativity, associativity, and distributivity. The typical examples are the ﬁeld of rational numbers (usually denoted Q ), the ﬁeld of real numbers R , and the ﬁeld of complex numbers C ; however as just mentioned not all examples are so familiar. The integers do not constitute a ﬁeld because in general it is not possible to divide one integer by another and have the result still be an integer. A ﬁeld is, by deﬁnition, a set F , say, equipped with two operations, + (addi- ﬁeld tion) and · (multiplication), which satisfy the following seven usual arithmetic axioms: (1) (Closure) For each a and b in F , the sum a + b and the product a · b are well-deﬁned members of F . (2) (Commutativity) For all a and b in F , a + b = b + a and a · b = b · a . (3) (Associativity) For all a , b , and c in F , ( a + b ) + c = a + ( b + c ) and ( a · b ) · c = a · ( b · c ). (4) (Distributivity) For all a , b , and c in F , a · ( b + c ) = a · b + a · c and ( a + b ) · c = a · c + b · c . (5) (Existence of identity elements) There are distinct elements 0 and 1 of F such that, for all a in F , a +0 = 0+ a = a and a · 1 = 1 · a = a . (6) (Existence of additive inverses) For each a of F there is an ele- ment - a of F such that a + ( - a ) = ( - a ) + a = 0. (7) (Existence of multiplicative inverses) For each a of F that does not equal 0, there is an element a - 1 of F such that a · ( a - 1 ) = ( a - 1 ) · a = 1. It should be emphasized that these common arithmetic assumptions are the only ones we make. In particular we make no ﬂat assumptions about operations

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A.1. BASIC ALGEBRA A-127 called “subtraction” or “division”. These operations are best thought of as the “undoing” respectively of addition and multiplication and, when desired, can be deﬁned using the known operations and their properties. Thus subtraction can be deﬁned by a - b = a +( - b ) using (6), and division deﬁned by a/b = a · ( b - 1 ) using (7) (provided b is not 0). Other familiar arithmetic properties that are not assumed as axioms either must be proven from the assumptions or may be false in certain ﬁelds. For instance, it is not assumed but can be proven that always in a ﬁeld ( - 1) · a = - a . (Try it!) A related, familiar result which can be proven for all ﬁelds F is that, given a and b in F , there is always a unique solution x in F to the equation a + x = b . On the other hand the properties of positive and/or negative numbers familiar from working in the rational ﬁeld Q and the real ﬁeld R do not have a place in the general theory of ﬁelds. Indeed there is no concept at
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BasicAlg - A-126 A.1 A.1.1 Basic Algebra Fields In doing...

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