For the learned mathematician it may be obvious and not worth mentioning: that the [fundamental theorems of arithmetic](https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic) and [algebra](https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra) look very similar and have to do with each other, in abbreviated form:

$$ n = p\_1\cdot p\_2 \cdots p\_k$$

$$ P(z) = z\_0\cdot(z\_1 -z)\cdot (z\_2 -z) \cdots (z\_k -z)$$

which makes obvious that the irreducible polynoms of first degree play the same role in $\mathbb{C}[X]$ as do the prime numbers in $\mathbb{Z}$ (which both are unitary rings). It also gives — in this special case — the wording "fundamental theorem" a specific meaning: It is stated that and how some irreducible elements build the fundaments of a structure.

Is this analogy helpful, or is it superficial and maybe misleading? If the former, can it be formalised? If the latter, what are the differences that make it merely superficial?