Let a(x) and b(x) be in F[x]. I need to prove that if a(x) and b(x) determine the same function, and if the number of elements in F exceeds the degree of a(x) as well as the degree of b(x), then a(x) = b(x).
I know what if a(x) and b(x) determine the same function means. It means that a(x) - b(x) gives you the zero function. I dont know what if the number of elements in F exceeds the degree of a(x) as well as the degree of b(x) means. I also do not see how either of the two can conclude a(x) = b(x). Can anyone help me?
The terminology is certainly confusing. I think what you mean by “zero function” is the constant function on whose image is 0, not the zero polynomial in . In other words, and determine the same function on F iff considered as functions .
An example to show what I mean. Suppose , , . Then, considered as functions , and are equal (to the identity function on ), but as polynomials in .
In the example, the number of elements of is not greater than the degree of . What you are asked to show is that if , then as polynomials in if as functions .