I don’t think you can make assumptions about how factors combine on multiplication (ie that multiplying by two adds one factor of two) in a context where intege

I don’t think you can make assumptions about how factors combine on multiplication (ie that multiplying by two adds one factor of two) in a context where integers can have rational noninteger roots. If we can have x=(a/b)^2 and y=(b/c)^2, then xy=(a/c)^2 so any factors of b^2 have somehow dropped out, rather than being added in to the factors of the product. Obviously in the end this all ends up contradicting the fundamental theorem of arithmetic, but it’s not clear this doesn’t wind up being circular.