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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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:
$$
k\cdot c(p) \cdot p''...
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Divisibility of a non invertible element in an integral domain
Let $A$ be a Unique factorization domain.
I want to prove :
Every non-invertible element of $A$ is divisible by at least one irreducible element.
My reason to believe that is because if $x$ is non ...
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Factorization of $x^3-10x+4$ over $\mathbb Q(\sqrt 2, \sqrt[3] 2)$
What is the factorization of $p(x) = x^3-10x+4$ over $ \mathbb Q(\sqrt 2, \sqrt[3] 2)?$
Since $p(x)$ is of degree $3$, it is irreducible over any field extension of $\mathbb Q$ if and only if no ...
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Do all polynomials $p(x)$ with integer coefficients output infinitely many composite numbers for integer $x$? [duplicate]
Inspired by this question, I was wondering if there was a simple proof that if $p(x)$ is a polynomial of degree $\geq 1$ and with integer coefficients, then $p(x)$ is a composite number for infinitely ...
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The diophantine equation $ a^2 + p b^2 = c^2 + p d^2 = q $ and related ring theory questions.
Inspired by norms of integral domains I started to think about the related diophantine equation for the norms.
Let $2 < p,q $ be given primes.
Consider the diophantine equation for positive ...
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Primes in quadratic rings that are not UFD's [duplicate]
In a quadratic ring $\mathbb{Z}[\sqrt{d}]$ that is not a UFD, is there a simple proof that if an element has a norm that is a prime integer, then that element of the ring is prime, not merely ...
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$\mathbb Z[x_n]$ is always a UFD?
Let $n$ be a positive integer and consider $f(x,n) = \frac{x^{(2n)}}{(2n)!}$ where $x^{(m)}$ is the rising factorial:
$$\begin{align*}
x^{(1)} &= x\\
x^{(2)} &= x(x+1)\\
x^{(3)} &= x(x+1)(...
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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....
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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. ...
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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. ...
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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 $f(X, Y)$ ...
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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}) ...
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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{...
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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 $...
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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 $...
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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?
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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 = (...
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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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Eigenspace decomposition for an integer matrix with integer eigenvalues
Suppose $A$ is a $n$ by $n$ matrix with integer entries and all eigenvalues $\lambda_1,\dots,\lambda_k$ of $A$ are also integers. If we consider the action of $A$ on $\mathbb{C}^n$, we know that $\...
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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] \...
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Proof that $v_{p}(ab)=v_{p}(a)+v_{p}(b)$
Recall.
We say that an integral domain $A$ is a unique factorization domain if the following statements are true :
(i) For every $a \in A \backslash \{0\}$, there exists a unit $u$, a non negative ...
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Content of polynomials in UFD.
Let $R$ be a UFD and $f(x)=\sum_{i=0}^{n} a_ix^i,g(x)=\sum_{j=0}^{m} b_jx^j\in R[x]$.The content of $f(x)g(x)$ is companion to the product of contents of $f(x)$ and $g(x)$. (By companion I mean for ...
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Finding a generator for the product of two ideals in $ \mathbb{C}[X][Y]$
Let $f, g\in\mathbb{C}[X,Y]$ be two irreducible elements in $\mathbb{C}[X,Y]$ such that $(f) \neq (g)$.
Give a generator for the ideal $((f) + (g)) \cdot ((f) \cap (g))$.
I figured out that $(f) \cap ...
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Viewing domain elements as fractions via natural map into fraction field (abuse of notation)
We have a simple proposition saying that
Proposition: Suppose $R$ is a UFD and let $Q(R)$ be the fraction field of $R$. Let $f(x)=\sum_{i=0}^n a_i x_i \in R[x]$ be a polynomial of degree $n\ge 1$. ...
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How to prove $x^p+y^p=\displaystyle{\prod_{k=0}^{p-1} (x+\xi_p^ky)}$
I am reading a "proof" of Fermat's last theorem which uses the assumption that $\mathbb{Z}[\xi_p]$ is a UFD (which is actually not true in general). This proof uses the following ...
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Questions about $K[x_1, \ldots , x_4]/(x_1x_2-x_3x_4)$ being a graded ring
So I was trying to understand this answer to the question of why $K[x_1, \ldots , x_4]/(x_1x_2-x_3x_4)$ ($K$ being a field) is not an UFD, and the author seems to use the fact that $K[x_1, \ldots , ...
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How to show the quadratic integer ring O is not a UFD.
Let $R=\mathbb{Z}[\sqrt{−n}]$ where $n$ is a squarefree integer greater than 3. Prove that $R$ is not a UFD. Conclude that the quadratic integer ring O is not a UFD for $D\equiv 2, 3$ mod $4$, $D < ...
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Problem with understanding why $\mathbb{C}[x, y] / (x^2 + y^2 - 1)$ is UFD
Reading this questions Ring of trigonometric functions with real coefficients
we can see that $\mathbb{C}[x, y] / (x^2 + y^2 - 1)$ is a UFD because it's isomorphic to Laurent polynomial ring which is ...
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Why is a semilocal domain $A$ a UFD, if its localisations at maximal ideals $A_{\mathfrak m}$ are UFDs?
This is Exercise 20.2 in Matsumura's Commutative Ring Theory:
Let $A$ be an integral domain. We say that $A$ is locally UFD if $A_{\mathfrak m}$ is a UFD for every maximal ideal $\mathfrak m$. If $A$ ...
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Is there are two prime elements which generate a proper ideal? [duplicate]
Let $A$ be a UFD, we know $A$ is a PID iff $A$ is a Bezout domain. Now I want to consider the case
$A$ is not a PID.
So the question is that does every UFD $A$ which is not PID admit two distinct ...
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Example of integral domain which is not a UFD [duplicate]
I was looking for an integral domain which is not a UFD and I've thought to
$\mathbb{C}[x,y]/(x-y^2)$
My questions are:
Is it right?
Is there any easier example?
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D.C.C condition of principal ideals on UFD
The question is:
Let $R$ be a UFD, and $I \neq (0)$ be an ideal of $R$. Prove that every descending
chain of principal ideals containing $I$ must stabilize.
Since for an UFD, A.C.C holds for principal ...
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Number of ring structures on $\mathbb{Z}$ with the usual multiplication
The usual addition is not the only one that makes $\mathbb{Z}$ into a ring with the usual multiplication.
Indeed, for any positive integer $n$, $\mathbb{Z}[x_1,x_2,...,x_n]$ is a unique factorization ...
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Is $\mathbb R[X,Y,Z]/(X^2+Y^2+Z^2+1)$ a unique factorization domain?
Let $\mathbb R$ be the field of real numbers. From Nagata's criterion for unique factorization domains, it follows that
$\frac{\mathbb R[X_1,\ldots,X_n]}{(X_1^2+\ldots+X_n^2)}$ is a unique ...
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Is it true that an uniformizing element of a discrete valuation ring $R$ is either zero or prime?
Let $R$ be a DVR. This means that it is a PID with a unique maximal ideal. Let us denote $m$ to be the maximal ideal. Then there is a unique irreducible element $\pi$ up to multiplication my a unit ...
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Problem in understanding of If $R$ is a UFD, then $R[x]$ is also a UFD(Gauss's theorem)
Gauss's Theorem:If R is a UFD then so is R[X], the ring of polynomials over R.
I am studying the the from this pdf. But I have problem in understanding.
Proof: Take a (non-zero) polynomial $f ∈ R[X]$....
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Minimal non-zero prime ideal in a UFD
I'm trying to understand the following Proposition:
Proposition: Let $R$ be a UFD (unique factorization domain) with subset $P$ of the set of irreducible elements of $R$. Then $P$ is in bijective ...
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Finding $x$ from $Ax = b$ with $QR$ factorization
I'm currently trying to find $x$ such as $A x = b$ where $A \in \mathbb{R}^{5\times 5}$, $x \in \mathbb{R}^{5}$ and $b \in \mathbb{R}^{5}$. I used for that the $QR$ factorisation. I know I have the ...
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Is $\mathbb{Z}[\sqrt[3]{2}]$ a UFD? [duplicate]
Is the quotient ring $\mathbb{Z}[X]/(X^3-2)$ a UFD?
I don’t have any idea to find whether some ring is a UFD or not.
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How can an irreducible polynomial be represented as a product of irreducible polynomials? [duplicate]
In Charles Pinter's Abstract Algebra, he presents the factorization theorem for polynomials. That is,
Every polynomial $a(x)$ of positive degree in $F[x]$ can be written as a product
\begin{equation*}...
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Show that the ring $\mathbb C[x^3,x^2y,xy^2,y^3]$ is normal, but it is not a unique factorisation domain
OExercise I am trying to solve:
Show that the ring $R = \mathbb{C}[x^{3}, x^{2}y, xy^{2}, y^{3}]$ is normal, but $R$ is not a unique factorisation domain.
Normal means that $R$ is integrally closed ...
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Let $R:=\mathbb{Z}[\sqrt{-6}]$. Prove that $R$ is not a Unique Factorization Domain
Question: Let $R:=\mathbb{Z}[\sqrt{-6}]$.
(a) Prove that $R$ is not a Unique Factorization Domain
(b) Find an irreducible element of $R$ that is not prime
(c) Find a non-principal ideal in $R$
(d) ...
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What are the integers that lack unique factorization in $\mathbb Z[\zeta_n],$ $n= 23$? [closed]
In the cyclotomic integers $\mathbb{Z}[\zeta_n]$, $n=23$ has class number $3$ and so unique factorization fails (https://en.wikipedia.org/wiki/Cyclotomic_field ). Can anyone give me examples of such ...
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