§ GCD is at most difference of numbers
- assume WLOG . Then, Let . Claim: .
- Proof: we have an by definition, hence we must have , and , are nonnegative. So .
- Intuition: the gcd represents the common roots of in Zariski land. That is, if are zero at a prime then so is .
- So, the GCD equally well represents the common roots of and .
- Now, if a number vanishes at a subset of the places where vanishes, we have (the prime factorization of contains all the prime factors of ).
- Since the GCD vanishes at the subset of the roots of , a subset of the roots of , and a subset of the roots of , it must be smaller than all of these.
- Thus, the GCD is at most .
- Why does GCD not vanish at exactly the roots of ? If and both take the same non-zero value at some prime then does too. But this is not a loacation where and vanish.