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?
Best Answer
In both cases, the theorem says "This ring is a unique factorisation domain and these are its irreducible elements". So in this sense, they are similar.
However, there are significant differences. In the case of $\Bbb Z$, the content is the unique factorization: since in any UFD, irreducible elements are prime, saying "and the irreducible elements are the prime numbers" doesn't add anything.
On the other hand, in the case of $\Bbb C[x]$, the content is what the irreducible elements are: given any field $K$, $K[x]$ is a UFD, and yet we know that the fundamental theorem of algebra doesn't hold over most fields (including $\Bbb R$, $\Bbb Q$, all finite fields, etc). So here the interesting part is that the polynomials of degree $1$ are the only irreducibles.