Questions tagged [unique-factorization-domains]

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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 ...
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5 votes
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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 ...
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2 votes
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$k[X_1,\ldots,X_n]/Q$ 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 ...
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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 ...
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6 votes
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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 \...
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7 votes
2 answers
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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 ...
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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 ...
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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 $\mathbb{Z}[\sqrt{-7}]\subset O_K$, ...
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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\...
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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 $A$...
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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}...
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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 ...
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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 $...
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4 votes
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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....
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9 votes
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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 ...
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20 votes
4 answers
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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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Unique factorization in polynomial rings

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, ...
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1 answer
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First-order UFD (factorial ring) condition / pre-Schreier rings

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 ...
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