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Question 1. When did the modern definition of a polynomial (as a sequence of coefficients, with multiplication defined by $\left(ab\right)_n = \sum\limits_{k=0}^n a_k b_{n-k}$) emerge?

Let me clarify: People have been working with some notion of polynomials for centuries, but mostly without a rigorous concept of what a polynomial is. From what I understand, "equations" rather than polynomials were the focus, and demarcating the precise extent of what an "equation" is was secondary to actually solving equations of given, fairly specific types. When the need for a rigorous definition became clear, it seems that the go-to solution was to define a polynomial as a polynomial function (e.g., §1 of Roland Weitzenböck, Invariantentheorie, 1923). This worked well for polynomials over $\mathbb R$ or $\mathbb C$, but not so much for polynomials over finite fields (which may be why E. H. Moore had to define $\mathbb{F}_{p^n}$ as a quotient ring of a polynomial ring over $\mathbb Z$ rather than of a polynomial ring over $\mathbb{F}_p$ in his 1893 talk).

What I call the "modern definition" of a polynomial is one of the following two definitions:

  • either as an infinite sequence $\left(a_0, a_1, a_2, \ldots\right)$ of coefficients (with all but finitely many $n$ satisfying $a_n = 0$), with addition defined entrywise and multiplication defined by the $\left(ab\right)_n = \sum\limits_{k=0}^n a_k b_{n-k}$ formula (and the indeterminate defined as the sequence $\left(0,1,0,0,0,\ldots\right)$ rather than being some mythic "symbol" or "indeterminate value" or "variable"),

  • or as the monoid ring of the additive monoid $\left\{0,1,2,\ldots\right\}$.

The second of these two definitions appears in §15 of B. L. van der Waerden, Moderne Algebra I, 2te Auflage 1937. (I can't check the 1st edition, as I don't have it.) This gives a lower bound on the age of the modern definition. As for an upper bound, I am perplexed. Nicolas Bourbaki's Elements of the History of Mathematics grazes the question on its page 82:

In all this research, the fields that arise consist of "concrete" elements in the sense of classical mathematics - (complex) numbers or functions of complex variables. 32 But already Kronecker, in 1882, takes full account of the fact (obscurely sensed by GaUBB and Galois) that "indeterminates" only play the role in his theory of the basis elements of an algebra, and not that of variables in the sense of Analysis ([186 a], v. II, pp. 339-340); and, in 1887, he develops this idea, linked to a vast programme that aims at nothing less than recasting all mathematics by rejecting all that cannot be reduced to algebraic operations on the integers. It is on this occasion that, taking up an idea of Cauchy ([56 a], (1), v. X, pp. 312 and 351) who had defined the field $\mathbb C$ of complex numbers as the field of remainders $\mathbb R\left[X\right]/\left(X^2+1\right)$, Kronecker shows how the theory of algebraic numbers is completely independent of the "fundamental theorem of algebra" and even of the theory of real numbers, all fields of algebraic numbers (of finite degree) being isomorphic to a field of remainders $\mathbb Q\left(X\right)/\left(f\right)$ ($f$ an irreducible polynomial over $\mathbb Q$)([186 a], v. 1111, pp. 211-240).

Following the references here, I see some signs of the modern definition slowly dawning in the 19th Century: [Note: "Polynomials" are often called "forms" before ca. the 1930s, particularly when they are homogeneous.]

So my question is: Who completed this definition? Weber (later)? Kronecker? Dedekind? Noether? van der Waerden? And while at that:

Question 2. Who defined formal power series?

Question 3. Who defined polynomials in noncommutative indeterminates?

These are litmus tests for Question 1; a rigorous definition of polynomials needs only a few trivial modifications to define formal power series, and a definition of noncommutative polynomials is not far away either.

The concept of a polynomial is, in a way, similar to the number $0$: It's an idea that, on its own, appears pedantic and meatless. But once it is established, it becomes a building block of a whole discipline that no one would want to be missing. (19th Century algebraists like Sylvester tend to use the unique factorization property of a polynomial ring without ever explaining what it is that they are factoring; from a modern perspective, this is a gaping hole in their proofs.) My hope is that, as this specific matter is much younger than the number $0$, we may have better clues to its authors.

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    $\begingroup$ What you call the "modern definition" was the original one given by al-Karaji and al-Samawal back in the 12th century. The latter wrote powers and the corresponding coefficients as columns of a table, and gave a multiplication rule equivalent to the convolution formula (al-Karaji's descriptions were verbal and more obscure). He even allowed negative powers (hence Laurent polynomials), and gave an algorithm for the "synthetic division". See e.g. Katz, History of mathematics, 9.3.3 $\endgroup$ – Conifold Jun 7 at 23:08
  • $\begingroup$ Interesting! This is pretty much the modern definition. I take it he didn't prove anything, though? (The question is nontrivial, because he had something very much resembling induction in his work, so it's not like the toolkit was entirely missing.) $\endgroup$ – darij grinberg Jun 8 at 10:21
  • $\begingroup$ Not in the modern sense, but that can be said about most of algebra until the late 18th century. Hudde further treated polynomials purely algebraically in the 17th. Dieudonne, before reinterpreting Euler's manipulations in terms of formal power series, states the following:"These series appear in print in Hilbert's Grundlagen der Geometrie, apparently for the first time". $\endgroup$ – Conifold Jun 8 at 10:46

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