Questions tagged [unique-factorization-domains]
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8
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\...
2
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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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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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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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How to compute a smooth number over a factor base in General Number Field Sieve (GNFS) factoring algorithm?
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) + ...
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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}...