Exercise · ZW-EUCLIDEAN-DOMAIN-007 · ★★☆ Intermediate
In \(\mathbb{Q}[x]\), compute the monic greatest common divisor of \(x^{3} - x^{2} + x - 1\) and \(x^{3} + x^{2} + x + 1\).
Choose one option.
Show answer
- A. \(x^{2} + 1\)
Explanation
We factor \(f=(x-1)(x^2+1)\) and \(g=(x+1)(x^2+1)\); the largest common factor is \(x^2+1\). With the algorithm: \(g-f=2x^2+2\) divides both.
Review the concept: Euclidean domain
An integral domain \(D\) with a norm \(N\) (a function from \(D\setminus\{0\}\) to the nonnegative integers) allowing division of \(a\) by \(b\neq0\) with quotient \(q\) and remainder \(r\) zero or of smaller norm than \(b\).
\[a=bq+r,\qquad r=0\ \text{ or }\ N(r)
More practice problems: euclidean domain
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- ZW-EUCLIDEAN-DOMAIN-008 If D is a Euclidean domain, what can be said about its ideals?
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