# Questions tagged [unique-factorization-domains]

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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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### 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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### 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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### 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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### 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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### 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 $A$...
### 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 ...