proof of Bezout’s Theorem

Let D be an integral domain with an Euclidean valuation. Let a,bD not both 0. Let (a,b)={ax+by|x,yD}. (a,b) is an ideal in D{0}. We choose d(a,b) such that μ(d) is the smallest positive value. Then (a,b) is generated by d and has the property d|a and d|b. Two elements x and y in D are associateMathworldPlanetmath if and only if μ(x)=μ(y). So d is unique up to a unit in D. Hence d is the greatest common divisorMathworldPlanetmathPlanetmath of a and b.

Title proof of Bezout’s Theorem
Canonical name ProofOfBezoutsTheorem
Date of creation 2013-03-22 13:19:58
Last modified on 2013-03-22 13:19:58
Owner Thomas Heye (1234)
Last modified by Thomas Heye (1234)
Numerical id 7
Author Thomas Heye (1234)
Entry type Proof
Classification msc 13F07