TITLE: Analogy between the fundamental theorems of arithmetic and algebra
QUESTION [21 upvotes]: For the learned mathematician it may be obvious and not worth mentioning: that the fundamental theorems of arithmetic and 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?

REPLY [12 votes]: Some other answers already make very good points. I just want to add that I think the truly amazing analogy is

"polynomials are the integers among the functions"; "polynomials behave like integers, and integers behave like polynomials"

which I could formally just state as: both $\Bbb Z$ and a polynomial ring $k[x]$ (over any field $k$) are Euclidean domains.
This realisation (that one can do division with remainder, hence has unique factorisation, what this means about the fraction field and its extensions, localisation, sheaves, ...) is indeed a profound insight, and arguably the analogy (and its generalisations) are a cornerstone of modern algebraic geometry and number theory.
That analogy, I think, should indeed be stressed more often. I sometimes mention it to my undergrad students, saying:

You learnt factoring numbers, then you learnt factoring polynomials, have you ever wondered what is the relation? Specifically, for numbers you end up with primes which you cannot factor anymore; are there polynomials like that? Which ones?
Or: In middle school you divided integers with remainder, in high school you divided polynomials with remainder. Well: the integer part of a fraction tells you its size, the polynomial part of a rational function tells you its behaviour for $x\to \infty$ ...
Or: Rational functions = quotients of polynomials, like rational numbers = quotients of integers. But they are not complete, limits of them are analytic or meromorphic functions (like $\sin, \tan$) -- just like rational numbers are not complete, limits of them are possibly transcendental numbers (like $\pi$) ...).

But this has nothing to do with $\Bbb C$. The Fundamental Theorem of Algebra, completely sidestepping that analogy, rather focusses on the fact (surely remarkable, but unrelated to all that) that in $\Bbb C[x]$ the "primes" are very easy.