Questions tagged [unique-factorization-domains]

A commutative ring with unity in which every nonzero, nonunit element can be written as a product of irreducible elements, and where such product is unique up to ordering and associates. Also called UFD or "factorial ring"

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Can a unique factorisation domain have a largest prime?

Suppose $R$ is a UFD and $(R,\leq)$ is an ordered ring. Is it possible that $R$ has a largest prime element? Below is my attempt so far to answer this myself, though I'm still unsure what the ...
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Irreducible elements, prime elements, prime ideals, and maximal ideals [duplicate]

I'm trying to get the four concepts listed in the title straight in my mind and elucidate the relationship between them. Could someone check the following statements and let me know if they are all ...
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Mistake in Proof "Every unique factorization domain is a principal ideal domain"

While doing my Algebra HW, I "proved" that every unique factorization domain (UFD) is a principal ideal domain (PID). I know that this is not true, however I fail to see where exactly is ...
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What property a ring extension of UFDs $R\subseteq S$ must have, such that for $a,b\in R$, $a | b$ in $S \implies a | b$ in $R$?

Let $R,S$ be two UFDs, and $R \subseteq S$, a ring extension with the following property (P) $$(\forall a,b\in R) (\ a|b \ in \ S \longrightarrow a|b \ in \ R).$$ Question: What are some conditions ...
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R is UFD. R is PID if every prime ideal is principal. [duplicate]

Suppose not. We consider the set of all non-principal ideals, $S$. Order $S$ by inclusion. We show S satisfies all the conditions in Zorn's Lemma. So it has a maximal element. If we show the maximal ...
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normalization of an algebraic curve

I am trying to solve this exercise: Let $f(x)$ be a polynomial with distinct roots different from $0$. What is the normalization of the algebraic curve $y^2=x^2f(x)$? My attempt: We call the base ...
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Are there rings where factorization is unique, but does not necessarily exist?

It feels like this should be a well-known question, but I can't find any related questions on this site by searching; apologies in advance if this is a duplicate. I assume rings are commutative with ...
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Explicit proof of the fact that a domain which is not a UFD is not a PID

In the same spirit as this question, I would like to prove explicitely that if $R$ is a domain which is not a UFD, then it is not a PID. I am interested in the case where there is an element $a\in R$ ...
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1 vote
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Need help understanding the proof showing the existence of gcd in an UFD.

Background: Definition: Let $a_1,a_2,\ldots a_n$ be elements (not all zero) of an integral doain $R.$ A $\textbf{greatest common divisor}$ of $a_1,a_2,\ldots, a_n$ is an element of $d$ of $R$ such ...
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Need help understanding the proof that if $R$ is a PID, every nonzero, nonunit elements of $R$ is a product of irreducibles.

Background: Lemma 10.9: Let $a$ and $b$ be elements of an integral domain $R.$ Then $(a)\subset (b)$ if and only if $b\mid a.$ $(a)=(b)$ if and only if $b\mid a$ and $a\mid b.$ $(a)\subsetneq (b)$...
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Explicit proof of the fact that a non integrally closed domain is not a UFD

Context. Let $R$ be a domain. It is well-known that $R$ is a PID $\Rightarrow$ $R$ is a UFD $\Rightarrow$ $R$ is integrally closed (in its field of fractions). In other words, we have $R$ is not ...
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Factorize $x^4 + 4x^2 +1$ over $\mathbb{F}_{29}$

How do I factorize $x^4 + 4x^2 +1$ over $\mathbb{F}_{29}$? I checked that the discriminant $D = 16 -4 = 12$ is not a square ($12^{14} = -1 \mod 29$) so this polynomial has no roots. Therefore it's ...
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Factorization into irreducibles in $\mathbb{Z}[i\sqrt{n}]$ with square-free $n \in \mathbb{Z}^{+}$

I'm trying to arrive at a proof that every nonzero $\alpha \in \mathbb{Z}[i\sqrt{n}]$ ($n \in \mathbb{Z}^{+}$ square-free) that is not an unit can be factored into the product of irreducible elements ...
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Divisibility of a non invertible element in an integral domain

Let $A$ be a Unique factorization domain. I want to prove : Every non-invertible element of $A$ is divisible by at least one irreducible element. My reason to believe that is because if $x$ is non ...
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Factorization of $x^3-10x+4$ over $\mathbb Q(\sqrt 2, \sqrt[3] 2)$

What is the factorization of $p(x) = x^3-10x+4$ over $\mathbb Q(\sqrt 2, \sqrt[3] 2)?$ Since $p(x)$ is of degree $3$, it is irreducible over any field extension of $\mathbb Q$ if and only if no ...
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The diophantine equation $a^2 + p b^2 = c^2 + p d^2 = q$ and related ring theory questions.

Inspired by norms of integral domains I started to think about the related diophantine equation for the norms. Let $2 < p,q$ be given primes. Consider the diophantine equation for positive ...
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Primes in quadratic rings that are not UFD's [duplicate]

In a quadratic ring $\mathbb{Z}[\sqrt{d}]$ that is not a UFD, is there a simple proof that if an element has a norm that is a prime integer, then that element of the ring is prime, not merely ...
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Bezout Lemma in $\mathbb{Z}[X]$

I'm thinking about the relation between the GCD, the resultant, and the Bezout lemma/identity for the ring $\mathbb{Z}[X]$. Take two elements $f,g \in \mathbb{Z}[X]$. We know: the GCD exists, because ...
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UFD implies GCD

Let $R$ be an integral domain. Assume $R$ is UFD, show $R$ is GCD. I did not find proof anywhere except one in proofwiki, but unfortunately there is a crucial part which I do not understand. https://...
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More examples that are elementary of GCD domains that are not UFD domains.

What are some more examples that are elementary of GCD domains that are not unique factorization domains? For my personal notes, I'm looking for examples of rings that belong to one rung but not the ...
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Unique Factoring in $\mathbb{Z}[x]$

While in class, my professor gave us a proposition that if $R$ is a unique factoring domain, then $R[x]$ is also a unique factoring domain. While in class I didn't think too much about it, but now I ...
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Proving that fractions can be written in lowest terms in a unique factorization domain [duplicate]

When proving that rational numbers can be written in their lowest terms, we used the well-ordering principle to choose a numerator that is the smallest and then it contradicts the fact that this ...
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Linear systems over UFDs

When Hermite and Smith normal form are discussed in the textbooks (e.g. the one of Adkins and Weintraub) they assume the ring to be a PID. What goes wrong over UFDs? Can one still do something similar ...
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Irreducibility using reduction mod $p$.
I know that if $R$ is a UFD and $\mathfrak{p} \subset R$ a prime ideal and $f$ a polynomial in $R[X]$ with content 1, s.t. $\deg(f)=\deg(\bar{f})$ then: $\bar{f}$ is irreducible in \$R/\mathfrak{p}[X] \...