Let $\mathfrak{D}$ be an integral domain with an identity, and let
$$f(x) = \sum_{k=0}^na_nx^k$$
be some non-constant in the polynomial domain $\mathfrak{D}[x]$. Are there any non-trivial theorems on or relationships between $g=\gcd(a_0,a_1\dots a_n)$ and $f(x)$? Just a shower thought, couldn't conclude anything other than basic in the ten minutes I played with it.
Yes, the coefficient gcd is one definition of the content $c(f)$ of a polynomial over a UFD or GCD domain. An important result about such that plays a fundamental role in divisibility theory in polynomial rings is
Gauss's Lemma $\,\ c(fg)\, =\, c(f) c(g)\ $
One way to generalize this to arbitrary commutative rings is to define the content to be the ideal generated by the coefficients. Then we have
Dedekind Mertens Lemma $\,\ c(f)^n c(fg)\, =\, c(f)^{n+1} c(g)\ $ for $\, n = \deg(g)\ \ ({\rm or\ \#terms}-1)$
See here for closely related versions (Dedekind's Prague theorem a.k.a. Gauss-Kronecker Lemma).
A comprehensive survey of this and related topics is D.D. Anderson: GCD domains, Gauss' lemma, and contents of polynomials, 2000.
Kronecker used polynomial contents as a foundation (alternative to Dedekind's ideal theory) for the study of factorization in algebraic number rings - what is known as divisor theory nowadays. For a modern introduction see the following (and search on "Kronecker function rings")
Friedemann Lucius. Rings with a theory of greatest common divisors.
manuscripta math. 95, 117-36 (1998).
Olaf Neumann. Was sollen und was sind Divisoren?
(What are divisors and what are they good for?) Math. Semesterber, 48, 2, 139-192 (2001).
