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Infinitesimal calculus and the introduction of derivatives is often linked to Newton and Leibniz.

I was wondering, when and why the idea of studying formal derivatives (e.g., of a formal polynomial) emerged?

For instance, what were the motivations to introduce derivatives to study things over finite fields and their extensions (where the visual intuition of infinitesimal as linked to Newton and Leibniz is lost)?

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  • $\begingroup$ Of course formal derivatives for finite fields is much more recent, since finite fields only began in the 1800s $\endgroup$ Mar 13 at 15:16
  • $\begingroup$ Finite fields aside, the idea of detecting double roots of polynomials by using (formal) derivatives predates infinitesimal calculus. It was introduced by Descartes in La Geometrie (1637) and developed by Hudde, see Is there a 'lost calculus'? A point of tangency can be interpreted as a double root of a system of a polynomial and a line equation, and the corresponding coefficient of the line equation is the derivative. In the algebraic case one does not need limits or infinitesimals even with visual intuition. $\endgroup$
    – Conifold
    Mar 13 at 17:05
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The motivation for applying derivatives to polynomials over general fields is their use in detecting multiple roots: if $K$ is a general field, a polynomial $f(x)$ in $K[x]$ has no repeated roots if and only if $f(x)$ is relatively prime to $f'(x)$ in $K[x]$.

Formal derivatives on polynomials in $K[x]$ for a general field $K$ were introduced by Steinitz in Section 9 of his big 1910 paper on the algebraic theory of fields and used in Section 10 in order to determine when a polynomial (especially an irreducible polynomial) over a general field has a multiple root. (Perfect fields are defined in Section 11 and separable field extensions are defined in Section 13 as extensions of the "first kind".)

Bourbaki writes in the Historical Note after the section Commutative Fields in his book Algebra II that in the 19th century, Dedekind "sensed" the general notion of a derivation (look at his paper here and see the footnote by Emmy Noether on the 2nd page) and that Galois "is fully aware of the formal character" of differentiation of polynomials and used this to show an irreducible polynomial in $\mathbf F_p[x]$ has no multiple roots in his development of finite fields. So if you are looking for the very first use of derivatives outside the setting of real or complex polynomials, then it goes back to Galois for the case of polynomials over a finite field.

Going back even farther, the use of derivatives as a formal operation (no limits) to detect multiple roots of real polynomials in some sense precedes calculus by Newton and Leibniz! This is called Hudde’s rule and you can read about it here.

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