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There are a few proofs in which the technique is to expand the product of some formal polynomials in $\mathbb{R}[x_1,x_2,\ldots,x_k]$ in more than one distinct way and then we can match up the coefficients. By formal, I mean that $x$ does not actually represent a number like in analysis, but it exists for algebraic book-keeping purposes. Some examples are:

  1. A proof of the Vandermonde identity
  2. A proof of the weakest variant of Wolstenholme's theorem
  3. Nathan Fine's proof of Lucas's theorem

Generally, these arguments use one of the following two facts for a polynomial $P$: \begin{align*} P^{ab}&=\left(P^{a}\right)^b,\\ P^{a+b}&= P^{a}P^{b}, \end{align*} though I am sure that there are other ways of expanding in two different ways. How do we know that coefficients can be matched up in this way?

I tried to prove it but I could not even formulate the general problem precisely. I suspect that this is a theorem in ring theory that is skipped over in most courses because of its techincal nature. Or perhaps this is a trivial issue, but I don't know. A proof or a reference would be appreciated.

Favst
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  • try to compute the coefficient of $x^c$ for every c in the two different ways and compare them – Exodd Aug 13 '20 at 22:37
  • or better, use the property that there is a unique polynomial of degree $<n$ such that on $1,2,\dots,n$ takes certain values – Exodd Aug 13 '20 at 22:38
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    The ring of formal polynomials has well-defined addition and multiplication. That is, the coefficients of a sum or product are uniquely determined. Thus, $P^aP^b=P^{a+b}$ just as in any ring with multiplication that is commutative and associative. – Somos Aug 13 '20 at 22:43
  • @Somos do you have a reference to a proof of that? I'm not doubting its truth, but I would appreciate being able to read what it takes to prove it. – Favst Aug 13 '20 at 22:45
  • Perhaps the answers to MSE question 216470 "What is a formal polynomial?" is enough proof for you. – Somos Aug 13 '20 at 22:51
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    Near-duplicate: https://math.stackexchange.com/questions/77671/how-to-show-associativity-of-multiplication-of-polynomials-in-rx-where-r – Eric Wofsey Aug 13 '20 at 23:25
  • You are asking for a proof of the fact that "formal polynomials" are indeed a ring. That is a perfectly legitimate question. It is answered (with better proofs in some books than in others) in pretty much every introductory book in abstract algebra. We wouldn't call it the "ring of polynomials" if it wasn't truly a ring, would we? –  Aug 14 '20 at 04:48

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