# Questions tagged [unique-factorization-domains]

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Following this on page 12, I understand the first steps of the general number field sieve (GNFS) algorithm for factoring as follows: Step 1: Let $$N = 77$$ and choose $$m = 4$$ Then $$N=77 = 1(4^3) + ... 13 votes 2 answers 383 views ### Tensor product of finite type UFD algebras over an algebraically closed field is again UFD? Let K be an algebraically closed field, A and B two finite type K-algebras which are assumed to be UFD. Is A \otimes_K B again a UFD? This question has been already asked here and here, but ... 1 vote 0 answers 108 views ### Square free elements in the value set of a polynomial At the end of this this article (conjecture 7.2), the author proposes (with some justification) a conjecture a bit more general than the following one: if P is a non constant and separable ... 5 votes 1 answer 236 views ### Is every matrix involution over a UFD diagonalisable? Let A be a UFD, that is also a k-algebra, where k is a field of characteristic \not=2 (for instance polynomials over k). Is every involution in \mathrm{GL}_n(A) diagonalisable? This is of ... 2 votes 1 answer 147 views ### k[X_1,\ldots,X_n]/Q is UFD for non-singular quadratic form Q and n\ge 5 I am looking for a reference for the following result. Thanks in advance. Let k be a field of any characteristic other than 2. Klein and Nagata showed that the ring R:=k[X_1,\ldots,X_n]/Q is a ... 3 votes 1 answer 223 views ### Do there exist irreducible elements in this domain? I asked this question on MSE. Here also I have the same motive in the question. Let D= \{\,a_1x^{r_1} + \cdots + a_n x^{r_n} \, \vert \, a_i \in \mathbb{C} \text{ for } i= 1,2,\dots,n \text{ and ... 6 votes 1 answer 260 views ### local UFD with dimension less than or equal 3 is catenary Let R be a commutative ring with identity. Then R is \textit{catenary} if for each pair of prime ideal p \subsetneq q, all maximal chains of prime ideals p = p_0 \subsetneq p_1 \subsetneq \... 7 votes 2 answers 549 views ### Zermelo's proof for unique factorisation In Peter Bundschuh's "Einführung in die Zahlentheorie" I came across a possibly well-known but to me rather peculiar proof of unique factorisation, which is attributed to Ernst Zermelo. The proof ... 1 vote 0 answers 250 views ### If R is UFD , then does R \cong R[X,Y] imply R \cong R[X]? R is isomorphic to R[X,Y], but not to R[X] shows that it is possible to have commutative ring R with unity such that R \cong R[X,Y] but R \ncong R[X]. My questions are: Is it possible ... 0 votes 1 answer 474 views ### Why is this not a non-unique factorization in the integer ring for \mathbb{Q}[\sqrt{-7}] when 7 is a Heegner? [closed] 7 is a Heegner number. Therefore the integer ring O_K corresponding to K=\mathbb{Q}[\sqrt{-7}] is a unique factorization domain. Now, it is easy to show that \mathbb{Z}[\sqrt{-7}]\subset O_K, ... 7 votes 0 answers 297 views ### Expressing quartic Dirichlet characters modulo primes p\equiv 1\bmod 4 with Legendre symbols Looking through some old notes of mine from two years ago I found some crude notes writing what amounted to the statement that for any prime p\equiv 1\bmod 4 one could express for any odd integer p\... 2 votes 1 answer 584 views ### Sandwich theorem for UFD's Let A \subseteq A[w] \subseteq C be three Noetherian integral domains, over a field k of characteristic zero, with A \subseteq C an algebraic ring extension (in particular, w algebraic over A... 0 votes 0 answers 254 views ### For which monic irreducible f \in \mathbb{C}[x,y][T], \mathbb{C}[x,y][T]/(f) is a UFD? Let f=f(T) \in \mathbb{C}[x,y][T] be a monic irreducible polynomial: f=T^n+a_{n-1}T^{n-1}+\cdots+a_1T+a_0, a_j \in \mathbb{C}[x,y], 0 \leq j \leq n-1. Denote B=\mathbb{C}[x,y,T]/(f)=\mathbb{C}... 2 votes 0 answers 99 views ### Characterizing all simple algebraic ring extensions of \mathbb{C}[x] having no prime elements Let w be an algebraic element over \mathbb{C}[x], with minimal polynomial f(t)=c_mt^m+\cdots+c_1t+c_0, c_i \in \mathbb{C}[x]. Is it possible to characterize (in terms of the c_j's) all ... 2 votes 0 answers 126 views ### Factorially closed subrings Lemma 3.2 says: Let A be a UFD. Let R \subseteq A be a subring of A such that R^* = A^*. The following conditions are equivalent: (i) Every irreducible element of R remains irreducible in ... 4 votes 1 answer 178 views ### Existence of Factor rings of UFDs which are UFDs Suppose that X=Spec(A) is an affine variety over an algebraically closed field k which is normal and such that Cl(X)=0. I am interested in hypersurfaces of X which again satisfy this condition.... 9 votes 0 answers 1k views ### When UFD implies PID The following result is too elementary, both to state and to prove, not to be known. Can someone give a reference? Is there any hope if you don't suppose UFD (i.e. move that from the hypothesis to ... 19 votes 6 answers 2k views ### Nonfree projective module over a regular UFD? What is the simplest example of a domain R which is regular (in particular Noetherian) and factorial which admits a finitely generated projective module that is not free? In fact I'll be at least ... 20 votes 4 answers 8k views ### For which c is \mathbb{Z}[\sqrt{c}] a unique factorization domain? a Euclidean domain? Let c be an integer, not necessarily positive and |c| not a square. Let \mathbb{Z}[\sqrt{c}] be the set of complex numbers$$a+b\sqrt{c}, \quad a, b\in \mathbb{Z}, which form a subring of the ...
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Everybody knows that polynomial rings over fields have unique factorization, and that if $R$ has unique factorization, then so does $R[X]$. And everybody knows who proved these results first. Well, ...
All rings in this post are commutative and with $1$. Everyone knows the definition of a factorial ring, a. k. a. unique factorization domain (UFD). I have been wondering about some variations ...