Question

Complete the proof of Corollary 10.4 by showing that an element d satisfying conditions (i) and (ii) is a greatest common divisor of a and b.

Short answer

It is shown that d is gcd of a and b

Step-by-step solution

Definition of gcd

Let R be Euclidean domain and $a,b\in R$(not both zero), a greatest common divisor of a and b is an element d such that

(i) d / a and d / b

(ii) If c / a and c / b,then $\delta \left(c\right)\le \delta \left(d\right)$

Proving that d is a greatest common divisor of a and b

Here, we have

(i) d / a and d / b

(ii) If c / a and c / b then c / d

Therefore, $c/d\Rightarrow d=cs\Rightarrow \delta \left(c\right)\le \delta \left(d\right)$

Therefore, we have

(i) d / a and d / b

(ii) If c / a and c / b then $\delta \left(c\right)\le \delta \left(d\right)$

Therefore, by definition of greatest common divisor, dis gcd of a and b.