UFD implies gcd
This article gives the statement and possibly, proof, of an implication relation between two integral domain properties. That is, it states that every integral domain satisfying the first integral domain property must also satisfy the second integral domain property
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Unique factorization domain
A gcd domain is an integral domain for which, given any two elements, there is an element, called the gcd of the two elements, such that any common divisor of the two elements divides the gcd.
- Pick two elements in the UFD. We need to exhibit an element which satisfies the property demanded of the gcd of these two elements
- Write down unique factorizations of both elements
- For each prime, pick as its gcd exponent the minimum of its exponents in the unique factorizations of both elements. The gcd is then the element where the exponent of each prime is its gcd exponent.