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2008-05-12T16:35:00Z

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{{integral domain property implication}}

==Statement==

===Property-theoretic statement===

The [[property of integral domains]] of being a [[unique factorization domain]] is stronger than the property of being a [[gcd domain]].

===Verbal statement===

Any [[unique factorization domain]] is a [[gcd domain]].

==Definitions used==

===Unique factorization domain===

A '''unique factorization domain''' is an [[integral domain]] where every element can be factorized uniquely as a product of [[irreducible element]]s.

===gcd 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.

==Proof==

===Proof outline===

* 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.

UFD implies gcd - Revision history

Vipul: 1 revision

Vipul at 20:44, 5 January 2008