Exercise 1873Algebra › Rings and fieldsL1Added 3 Nov 2026

GCD in $\mathbb{R}[X]$

#GCD#Euclidean algorithm#Bézout

Statement

Compute $\gcd(X^3-1,X^2-1)$ in $\mathbb{R}[X]$ , and find $U,V$ with $U(X)(X^3-1)+V(X)(X^2-1)=\gcd$ .

Hint

Euclidean algorithm: divide $X^3-1$ by $X^2-1$ .

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Division. $X^3-1=(X^2-1)X+(X-1)$ .

Division. $X^2-1=(X-1)(X+1)+0$ .

GCD. $X-1$ .

Bézout. $X-1=(X^3-1)-X(X^2-1)$ . So $U=-X$ , $V=1$ .

Conclusion. $\gcd=X-1$ , and $-X(X^3-1)+1(X^2-1)=X-1$ .