# 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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### 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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### $k[X_1,\ldots,X_n]/Q$ 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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### 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 ... 413 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 $\mathbb{Z}[\sqrt{-7}]\subset O_K$, ...
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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 ...