
To explain:
The integers set is a field or not.

Answer to Problem 35WE
The integers set is not a field.
Explanation of Solution
Definition of property of reciprocal:
A number x and its inverse or multiplicative inverse is the number that gives 1 when multiplied by x .
The product of a number and its reciprocal number is therefore 1 . This property is referred to as a reciprocal property.
Thus, from the above definition of the property of reciprocals for any real number a except 0 , there is a unique real Number 1a such that a⋅1a=1 .
2⋅12=1
The number 12 is not an integer.
We know that to satisfy the property of reciprocals an integer number is to be multiplied with its reciprocal. For all numbers except 1 , the reciprocal is a fraction.
Since for all numbers except 1 the reciprocal is a fraction, the integers set is not a field.
Chapter 1 Solutions
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