Can Redefining Operations Make Integers a Field?

[mruncleramos]

This problem comes from Halmos's Finite Dimensional Vector Spaces. Given that we can re-define addition or multiplication or both, is the set of all nonnegative integers a field? What about the integers? My thinking is that since the Rational numbers form a field, and they are countable, we can assigen each number of the aforementioned set a rational and then we can have a field with the integers representing rationals. Am I wrong? Edit: On second thought, this doesn't seem to right. Uniqueness is violated somewhere i think.

 

[AKG]

Technically speaking, a set is not a field, a set together with an addition and a multiplication, (F, +, *) is a field. Since the positive integers, non-negative integers, integers, and rationals are all countable (since there is a bijection between any pair of those sets), and since the rationals are a field, yes, you could make a field using anyone of those sets, you'd essentially just be relabelling the rationals in such a way that makes addition and multiplication look very strange. Really, if you could name infinitely many animals, then you can make a field of animals, but what the heck does "elephant + zebra = giraffe" mean?

 

[M98Ranger]

Your thinking is correct, the set of all nonnegative integers is not a field. In order for a set to be a field, it must satisfy certain properties such as closure under addition and multiplication, existence of additive and multiplicative inverses, and commutativity and associativity of operations.

While it is possible to assign each nonnegative integer a rational number and create a field, this does not satisfy the property of closure under addition and multiplication. For example, if we assign the nonnegative integer 2 the rational number 1/2, then 2+2=4, but 1/2+1/2=1, which is not a nonnegative integer. This violates the closure property and therefore, the set of nonnegative integers cannot form a field.

Similarly, the set of integers also does not form a field. While it satisfies the closure property under addition and multiplication, it does not have multiplicative inverses for all elements. For example, the integer 2 does not have a multiplicative inverse in the set of integers. This means that we cannot find another integer that when multiplied by 2 gives us the multiplicative identity, which is 1. Therefore, the set of integers does not satisfy all the properties required to be a field.

In summary, in order for a set to be a field, it must satisfy all the properties required for a field, and neither the set of nonnegative integers nor the set of integers satisfy all these properties.