# Questions tagged [unique-factorization-domains]

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questions with no upvoted or accepted answers

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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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### 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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### 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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### If R is UFD , then does R≅R[X,Y] imply R≅R[X]?

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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≅R[X,Y] but R≆R[X].
My questions are: Is it possible to have an example of a ...

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182 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 ...

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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}...