What is the prime factorization of zero?

Fundamental theorem of arithmetic¹: every whole number can be split into a product of prime numbers in a unique way.

Examples:

What is prime factorization of one? An empty sequence, because it is the multiplicative identity and an empty product is the identity.

Now given this theorem, let’s think of 6 from a different POV:

Then dividing by a prime p is equivalent to dropping a term of value p from the sequence on the right hand side.

Now let’s take a bold step and declare 0 has a factorization. Name it $F$(actorization).

But $\frac{0}{2}=0$, so dropping a term from $F$ should give us the factorization of zero. Which is $F$ itself!

This dropping property holds for any prime $p$, and by extension any nonzero number at all².

So $F$:

1. Has to be infinite length, else dropping a term from it would strictly shorten it.
2. Contains every prime infinitely many times since 0 can be divided arbitrarily many times without changing.

So the prime factorization of 0 is all the prime numbers, repeated a infinite number of times. Unlike every other standard natural number, not a finite sequence.

So if you take every prime and multiply them all by each other infinitely many times, you get zero. Trippy.