Actually, you're pretty much spot on. The word you want is "irreducible", rather than "indivisible". In general rings there are irreducible elements and prime elements, and they have different definitions. You're looking for a unique factorisation into irreducibles.

https://en.wikipedia.org/wiki/Irreducible_element https://en.wikipedia.org/wiki/Prime_element

Thanks!

Trying to read that is like trying to learn an entire new language by reading a sentence in it. Way too many words to already understand what it's even defining (commutative ring, irreducible polynomials, UFDs, principle ideal, nozero prime ideal... and I'm not following why p divides ab in R... or even exactly what that means).

Searching youtube kahn academy and numberphile, but not turning up anything. exactly how high level is this?

This is usually part of an introductory undergrad course in algebra. What you're looking for is called "ring theory".

Michael Artin's Algebra is a good, concrete book with tons of motivation and zero prerequisites. Wanna try it? ;)

I'm currently taking a course on this material in 2nd year undergrad.

It could also be false if there were two different pairs of prime numbers with the same product. See Answer 4 in the article.

Ah

I love it when my error is so simple that there isn't any question in my mind I missed something.

Problem is, we only ever mess about with small numbers. Take some absolutely humungous number, N, and find, somehow, that it's a times b. Suppose someone else asserts that it's c times d. These numbers are enormous, absolutely enormous, so the question is: if a, b, c, and d are all prime, is it really obvious that these two ways of writing N have to be the same?

Edit: As pointed out elsewhere (and in the sibling to this comment) there are systems where "prime" and "irreducible" are not the same thing. In the integers they are, and that's sort-of why the FTA is true, but in other places they aren't, which is why the FTA is not obvious.