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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 ...
Damalone's user avatar
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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 ...
Cezar's user avatar
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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 ...
Dwaipayan Sharma's user avatar
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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 $a\in R$ ...
GreginGre's user avatar
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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 $a\mid b.$ $(a)\subsetneq (b)$...
Seth's user avatar
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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 ...
GreginGre's user avatar
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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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Unique factorization domain property in $\mathbb{Z[\mathcal{i}]}$

Acknowledging that $\mathbb{Z[\mathcal{i}]}$ is a UFD, show that $11 + 14i, 23 − 12i, −91 + 230i, 305 + 192i$ cannot all be prime. By direct calculation, I checked that $$(11+14i)(305+192i)=(23-12i)(-...
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Let $R$ be a UFD, and let $S$ be a multiplicative subset of $R$ containing the unity of $R$ then $R_S$ is also UFD

Let $R$ be a UFD, and let $S$ be a multiplicative subset of $R$ containing the unity of $R$ then $R_S$ is also UFD. There are several answer on M.SE like here, here. But none of them resolve the ...
N00BMaster's user avatar
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Let R be a UFD such that each maximal ideal of R is a principal ideal, prove that R is a PID

I need help to the following problem: Problem: Let $R$ be a Unique Factorization Domain such that each maximal ideal of $R$ is a principal ideal. Then $R$ is a Principal Ideal Domain. Solution(my ...
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Proof about gcd and resultant in UFD

Let $R$ be a UFD and $f,g \in R[x]$ not both zero. Prove that $$\gcd(f,g) \in R[x] \backslash R \iff res(f,g)=0$$ The hint is to consider $R$ as embedded in its quotient field and rewrite the ...
Magne Seier's user avatar
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Primitive $g\mid af\Rightarrow g\mid f,\,$ for $\,0\!\neq\! a\in $ UFD A, $\,f,g\in A[x],\,$ by Gauss's Lemma

I'm studying Unique Factorization Domain (UFD) and there's a lemma in my textbook stating that: In polynomial ring $A[x]$ with $A$ is a UFD, if $g$ is a primitive polynomial and divides $af$ ($a\in A$...
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If every irreducible elements in Noetherian ring is prime, it is UFD

I have problem with the proof of theorem 4 in this link It says if $R$ is an Noetherian ring, we construct $\mathbb U$, the set of ideals generated by each element of $R$ that cannot be written as a ...
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Exhibit all the ideals in the ring $F[x]/(p(x))$, where $F$ is a field and $p(x)∈F[x]$ (describe them in terms of the factorization of $p(x)$)

This is Exercise 9.2.5 in Dummit and Foote's Abstract Algebra Exhibit all the ideals in the ring $F[x]/(p(x))$, where $F$ is a field and $p(x)$ is a polynomial in $F[x]$ (describe them in terms of ...
Stanarth's user avatar
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How can I compare the number of irreducible factors of one element with another?

Let $R$ an UFD and $a,b,c \in R$ be such that $a$ and $b$ are nonzero and noninvertible. Suppose $a = bc$. How can I show that the number of irreducible factors of $a$ is greater than or equal to the ...
Juan Herrera's user avatar
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What is the factorization of 26 in $\mathbb Z[i]$ into irreducible?

Here is the question I am trying to solve: What is the factorization of 26 in $\mathbb Z[i]$ into irreducible? My idea is: 26 = (5-i)(5+i) but it seems like this is an incomplete answer. Could anyone ...
Emptymind's user avatar
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If $R$ is not a UFD then $RS^{-1}$ is not a field

If $R$ is not a UFD then $RS^{-1}$ is not a field considering the set in $(*)$. Is this proof I did correct? Proof: Let $R$ integral domain, $P$ the set of all non-zero prime elements of $R$ and let $...
Juan Herrera's user avatar
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$x^2 + y^2 + z^2 - 2xy - 2xz - 2yz$ is irreducible over $\mathbb{Z}[x, y, z]$

I am trying to prove that $f(x,y,z) = x^2 + y^2 + z^2 - 2xy - 2xz - 2yz$ is irreducible over $\mathbb{Z}[x, y, z]$ (I am sure it is) and maybe over $\mathbb{R}[x, y, z]$ (almost sure). I checked that ...
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Deducing that $\mathbb Z[2 \sqrt{2}]$ is not a $UFD.$

Here is the question I am trying to understand the solution of the last part in it: Let $R$ be an integral domain with quotient field $F$ and let $p(x)$ be a monic polynomial in $R[x].$ Assume that $p(...
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Why $y$ is a rational integer?

I was reading that solution of $\mathbb{Z}[\sqrt{−n}]$ is not a UFD here Conclude that $\mathbb{Z}[\sqrt{−n}]$ is not a UFD. But I do not understand this line "So either $d = 1, 2$ or $\gcd(y,\...
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For compact and connected $K\subset\mathbb{C}$, $H(K)$ is a PID?

For compact and connected $K\subset\mathbb{C}$, let $H(K)$ be the set of functions that extend to a holomorphic function over an open neighborhood of $K$. This answer states that $H(S^1)$ is a UFD (...
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$c(kp) = kc(p)$ where $c$ is the content of a polynomial, $p$ a polynomial in $R[x]$ and $k \in R-\{0\}$ (without assuming Gauss' Lemma)

I have been trying to prove (without assuming Gauss' Lemma) that for $R$ a UFD and $p(x) \in R[x]$ then for $k \in R- \{0\}$ we have $c(kp) = kc(p)$. My approach is as follows: $$ k\cdot c(p) \cdot p''...
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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 ...
Laurent Claessens's user avatar
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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 ...
mick's user avatar
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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 ...
akay's user avatar
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$\mathbb Z[x_n]$ is always a UFD?

Let $n$ be a positive integer and consider $f(x,n) = \frac{x^{(2n)}}{(2n)!}$ where $x^{(m)}$ is the rising factorial: $$\begin{align*} x^{(1)} &= x\\ x^{(2)} &= x(x+1)\\ x^{(3)} &= x(x+1)(...
mick's user avatar
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2 votes
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Proving existence of a lcm in a Unique Factorization Domain $D$ using the ascending chain condition?

I'm aware there are many posts asking this question, but I was wondering if it was possible to prove this fact using ascending chains? The problem (as in the title) is the following: Let $D$ be a UFD....
Isochron's user avatar
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Is there a general formula for factorizing multivariate complex polynomials?

Let $\mathbb{C}[X_1,...,X_d]$ denote the space of polynomials over $\mathbb{C}$ in $d$ variables. In the case $d = 1$, we can always factorize a polynomial using its zeros and their multiplicity, i.e. ...
Andreas132's user avatar
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Characterization of irreducible multivariate polynomials over the complex numbers

Let $\mathbb{C}[X_1,...,X_d]$ denote the set of polynomials over $\mathbb{C}$ in $d$ variables. If $d=1$, we know that the irreducible polynomials are exactly the polynomials of degree $1$, i.e. ...
Andreas132's user avatar
3 votes
1 answer
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$A[X, Y ]/(X^2+Y^2−1)$ is integral domain

Deduce that if $A$ is a unique factorization domain (UFD) of char- acteristic zero, then $A[X, Y ]/(X^2+Y^2−1)$ is an integral domain. Let $B = A[X, Y]/(X^2 + Y^2 - 1)$, and suppose that $f(X, Y)$ ...
Abcd's user avatar
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Multiplicative Subsets of the Natural Numbers and Unique Factorization

Elementary number theory books often give as an example of non-unique factorization the set $S = \{4k+1: k \in \mathbb{N}\}$. $S$ doesn't have unique factorization because $(3 \cdot{7})(11 \cdot {19}) ...
Pete Klimek's user avatar
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2 answers
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Prove that $\mathbb{C}[x,y,z]/(y^2z-x^3+xz^2)$ is not isomorphic to $\mathbb{C}[x]$

I want to prove that $\mathbb{C}[x,y,z]/(y^2z-x^3+xz^2)$ is not isomorphic to $\mathbb{C}[x]$. My attempt is to see that because $\mathbb{C}$ is UFD, then $\mathbb{C}[x]$ is UFD. However, in $\mathbb{...
Jessen William's user avatar
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Decomposition of bivariate polynomials over finite fields as a sum of univariate products

Let $p$ be a prime. Given a bivariate polynomial $f(X,Y)\in \mathbb{F}_p[X,Y]$ with degrees $d_1,d_2$ in $X,Y$ respectively, what is the lowest known upper bound on the smallest integer $k$ such that $...
Mathdropout's user avatar
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When is $\dfrac{k[x,y,z]}{(x^a +y^b+z^c)} $ a unique factorization domain?

Let $k$ be an algebraically closed field of characteristic zero. Let $a,b,c$ be relatively prime positive integers. Then, is $\dfrac{k[x,y,z]}{(x^a +y^b+z^c)} $ a UFD? This question is motivated from $...
strat's user avatar
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Examples of Gorenstein local unique factorization domain of dimension $2$ and embedding dimension $4$

I am looking for an example of a local UFD $(R,\mathfrak m)$ of dimension $2$ which is also Gorenstein and $\mathfrak m$ is minimally generated by four elements. Does there exist any such examples?
Snake Eyes's user avatar
1 vote
1 answer
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Generator of ideal in $Q(\zeta_5)$

I am thinking about the splitting of 11 in $Q(\zeta_5)$. This totally splits, and I was told that since the minimal polynomial of $\zeta$ completely factorizes mod 11 as $$ x^4 + x^3 + x^2 + x + 1 = (...
Three aggies's user avatar
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2 votes
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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 ...
Johann Birnick's user avatar
4 votes
3 answers
523 views

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://...
Ziqiang Cui's user avatar
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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 ...
Greg Nisbet's user avatar
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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 ...
Ty Perkins's user avatar
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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 ...
AMX's user avatar
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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 ...
HCH's user avatar
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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] \...
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