San José State University

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The Definition and Structure
of a Mathematical Field

A field F consists of a set S of elements and two binary funcions f and g defined on S and whose values are in S. Both binary functions are associative, f(a, f(b,c) is equal to f( f(a, b), c) for all a, b and c in S and likewise for g. The binary function f, called addition is necessarily commutative; i.e., f(a, b) is equal to f(b,a) for all a and b in S. The other function g, called multiplication, is not necessarily commutative. There is an identity for each functions; i.e. there exist e such that f(e, a)=a for all a in S and an l such that g(l, a)=a for all a in S. There exist additive inverses for all elements of S and multiplicative inverses for all elements of S except the additive identity. This means that for any element a in S there exist a b such that f(a, b)=e. Such a b is denoted as −a. For any element a in S that is not e there exists a b such that g(a, b) is equal to l. The element b is usually denoted as a−1.

Furthermore there is distributivity of multiplication with respect to addition; i.e., g(a,f(b,c)) is equal to f((g(a,b), g(a,c)) for all a, b and c in S.

Among the algebraic structures similar to fields, fields are at the core of the system. This is represented as:

Commutative rings ⊃ integral domains ⊃ integrally closed domains
⊃ unique factorization domains ⊃ principal ideal domains ⊃
Euclidean domains ⊃ fields

Examples

The most important mathematical field for human endeavor is the field of the real numbers. The complex numbers also constitute a field.

Another set of fields is the sets of integers with the operations defined in terms of the remainders upon division by a prime number.

Consider the set of integers {0, 1, 2} with addition and multiplication modulo 3. Thus the multiplication table is




      0     1    2

0     0     0    0

1     0     1    2

 2     0     2    1  



Thus the multiplicative inverse of 1 and the multiplicative inverse of 2 is 2. "1" is the multiplicative identity and "0" the additive identity. The table of addition is



      0     1     2

0     0     1     2
	
1     1     2     0

2     2     0     1


(To be continued.)

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