Like all mathematics, the first thing to do is set the landsape in which you wish to work in. Let be a field and let be the commutative polynomial ring with variables (if you are familiar with this ring I would skip the next paragraph)
What do the elements of look like? Taking it is not hard to come up with elements, for example . One can quite easily start conjuring up polynomials for a given but a more interesting approach to answer the question, can we find a ‘generating’ set for our polynomials? This brings one to the notion of a moninomial which is simply a product of the form . We can simplify the notation by starting with an -tuple of non-negative integers and letting . Clearly a poynomial can be written as a finite linear combination of moninomials with the general form where and with also being finite. Without knowing what a polynomial is aprior this is the method used to define what a polynomial is with variables.
is a ring with addition and multiplicative defined how you would expect and as is a field the ring is commutative. Given we can think of it as an algebraic element of our ring or a function by swapping with some for all . This gives us two notions of zero; the zero element of and a zero function from .
To emphasise the difference let us take , and look at the polynomials and in . As algebraic elements clearly is the zero element and isn’t but as functions and so we can conclude both and are zero functions. To wrap up we have two distinct elements that are both zero functions .
This invokes a big question and one in which we will answer: When is the zero polynomial the only zero function? (I didn’t lead you with false pretenses by the title). The answer depends on the cardinality of .
If is finite of size by Lagrange’s Theorem and looking at the multiplicative group one can show for all so the polynomial is a zero function but is distinct from the zero element in . If we can define . This tells us that when is finite we can always come up with elements in that are non-zero and are a zero function so there is no unique zero function.
What about the case when is infinite? Well…
Theorem: Let be an infinite field, let . Then, if and only if is the zero function.
Proof of Theorem: Let us begin with the ‘easy’ direction, suppose trivially we must have for all . The other direction requires us to show that if for some such that for all then is the zero element in .
We will use induction on the number of variabels . When a non-zero polynomial in of degree has at most roots. Suppose such that for all and since is infinite must have infintely many roots and as the degree of must be finite we must conclude that .
Now assuming the inductive hypothesis for let be a polynomial such that for all . By collect the various powers of we can write in the form of where is the highest power of that appears in and . If we show that is the zero polynomial in variables this will force in .
If we fix we get a polynomial and by our hypothesis this polynomial vanishes for all . Using our above representation of we see the coefficients of are and hence for all . As is chosen arbitraty is the zero function and so by the inductive hypothesis is the zero polynomial in .
Bringing in both of these results lets us conclude that if a polynomial zero function is the zero element of then we must have that the field is infinite. I find this quite a remarkable result in the fact that knowledge about a zero function gives you an indication on the cardinality of .
The theorem we have just proved also lets us conclude that when is infinite polynomials are unique functions or more precisely for in if and only if and are the same functions. I leave the proof of this for the reader and as a hint consider the polynomial .