In field theory, we can define a function $d:F[X]\to F[X]$ where $\sum_{i=0}^n a_iX^i \mapsto \sum_{i=1}^n ia_iX^{i-1}$, where $F[X]$ is some polynomial ring.
Letting $d(f)=f'$ for $f\in F[X]$, I have recently learned in my Galois theory class that this function is useful in the following way:
- $f$ has no multiple roots in any extension field $\iff$ GCD$(f,f')=1$.
- If $f$ irreducible, then we have $f$ separable $\iff$ $f'\neq 0$.
Alternatively, we learn in calculus that we can calculate the derivative of a polynomial to get an equation that gives the slope of the tangent line at any point in the domain.
On the surface, this function feels totally unrelated to slopes or tangent lines, so it feels like a remarkable coincidence that their definitions coincide on polynomials. Is there any reason I should've expected a priori that they are the same? Is there some connection here I am missing? Or is it simply that those working in with these field extensions were aware of the tools of calculus and applied them here?
Thanks in advance for any thoughts on this.