I think you're mincing words here. I wasn't implying you were a layman to mathematics. Just that the definition you might have been basing that statement on was a laymen's definition. I know, from years of experience dealing with you that you have an awesome grasp of mathematics and computer science. And probably much more.
I didn't "hear" a definition of prime
Actually it's the Fundamental Theorem of Arithmetic, which is what I was referring to. The Fundamental Theorem of Algebra is something very different.
(Fundamental Theorem of Algebra)
Except that this modern definition, with which I'm familiar, is a consequence of its equivalence to the definition that satisfies the fundamental theorem of arithmetic and a desire to merge abstract algebra with number theory.
The modern definition of prime is an element of a commutative ring that is not 0, not a unit and generates a prime ideal (that's the condition I just described in my previous post). The "not a unit" excludes 1. But this is a convention that people have found convenient, and has no deeper meaning.
Yep. But this isn't really related to what we're talking about, which is the definition and whether 1's exclusion is a convention or a consequence of something deeper. Not the consequences of things being different.
If we were to consider 1 a prime, many theorems would have to be reworded to exclude it. But some other theorems would be simplified. For instance, see Wilson's theorem. There are also theorems that need to exclude 2 (example), but that doesn't mean that 2 isn't prime.