query: Does a UFD imply Euclidean Domain?
positive_passages.docid: 22789
positive_passages.text: How to determine whether a unique factorization domain is a principal ideal domain?
negative_passages.docid: 27600
negative_passages.text: Gödel's (in)completeness theorems and the axiomatization of Euclidean geometry
negative_passages.docid: 102914
negative_passages.text: Integral domain that is not a factorization domain
negative_passages.docid: 27652
negative_passages.text: metric on the Euclidean Group
negative_passages.docid: 141078
negative_passages.text: Domain over which a polynomial is strictly positive
negative_passages.docid: 42361
negative_passages.text: Why do we use the Euclidean metric on $\mathbb{R}^2$?
negative_passages.docid: 82464
negative_passages.text: $U^*\otimes V$ versus $L(U,V)$ for infinite dimensional spaces
negative_passages.docid: 83798
negative_passages.text: Does $A$ a UFD imply that  $A[T]$ is also a UFD?
negative_passages.docid: 50605
negative_passages.text: Extended Euclidean algorithm
negative_passages.docid: 9748
negative_passages.text: Maximum modulus principle for an unbounded function in the half right plane
negative_passages.docid: 75899
negative_passages.text: Principal ideal domain with finitely many ideals
negative_passages.docid: 67699
negative_passages.text: Integral domain that is integrally closed, one-dimensional and not noetherian
negative_passages.docid: 148804
negative_passages.text: Norm-Euclidean rings?
negative_passages.docid: 18484
negative_passages.text: Show that $k[x,y,z]/(xz-y^2)$ is not a UFD.
negative_passages.docid: 512
negative_passages.text: Finding generator for the ideal generated by $a$ and $b$ in an euclidean domain
negative_passages.docid: 24653
negative_passages.text: Integral in $n-$dimensional euclidean space
negative_passages.docid: 77877
negative_passages.text: Is there a domain "larger" than (i.e., a supserset of) the complex number domain?