Exercise · ZW-EUCLIDEAN-DOMAIN-009 · ★★★ Advanced
To prove that an ideal \(I\neq0\) of a Euclidean domain is principal, which element is taken as generator?
Choose one option.
Show answer
- C. A nonzero element of \(I\) with minimal norm
Explanation
If \(d\) has minimal norm and \(a\in I\), divide \(a=dq+r\). Then \(r=a-dq\in I\) and, if nonzero, it would have smaller norm than \(d\), absurd. So \(a\in(d)\). A maximal norm does not even exist in general.
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
- ZW-EUCLIDEAN-DOMAIN-001 In ℤ with N(n)=|n|, which quotient q and remainder r satisfy 17=5q+r with 0≤ r<5?
- ZW-EUCLIDEAN-DOMAIN-002 Which function works as a Euclidean norm on F[x], with F a field?
- ZW-EUCLIDEAN-DOMAIN-003 In the Gaussian integers ℤ[i] the norm N(a+bi)=a^2+b^2 is used. What is N(3+4i)?
- ZW-EUCLIDEAN-DOMAIN-004 In ℤ, use the Euclidean algorithm to compute gcd(252,198).
- ZW-EUCLIDEAN-DOMAIN-005 In ℚ[x], what is the greatest common divisor of x^3-1 and x^2-1?
- ZW-EUCLIDEAN-DOMAIN-006 In ℤ[i] with N(a+bi)=a^2+b^2, which is a valid division of 7+3i by 2 (that is, 7+3i=2q+r with r=0 o…
- ZW-EUCLIDEAN-DOMAIN-007 In ℚ[x], compute the monic greatest common divisor of x^3 - x^2 + x - 1 and x^3 + x^2 + x + 1.
- ZW-EUCLIDEAN-DOMAIN-008 If D is a Euclidean domain, what can be said about its ideals?