# proof of Bezout’s Theorem

Let $D$ be an integral domain with an Euclidean valuation. Let $a,b\in D$ not both 0. Let $(a,b)=\{ax+by|x,y\in D\}$. $(a,b)$ is an ideal in $D\neq\{0\}$. We choose $d\in(a,b)$ such that $\mu(d)$ 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 associate if and only if $\mu(x)=\mu(y)$. So $d$ is unique up to a unit in $D$. Hence $d$ is the greatest common divisor of $a$ and $b$.

Title proof of Bezout’s Theorem ProofOfBezoutsTheorem 2013-03-22 13:19:58 2013-03-22 13:19:58 Thomas Heye (1234) Thomas Heye (1234) 7 Thomas Heye (1234) Proof msc 13F07