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 is $\mathbb{Z}[\sqrt{-n}], n\ge 3$ not a UFD?

I'm considering the ring $\mathbb{Z}[\sqrt{-n}]$, where $n\ge 3$ and square free. I want to see why it's not a UFD. I defined a norm for the ring by $|a+b\sqrt{-n}|=a^2+nb^2$. Using this I was able to show that $2$, $\sqrt{-n}$ and $1+\sqrt{-n}$…
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Ring of integers is a PID but not a Euclidean domain

I have noticed that to prove fields like $\mathbb{Q}(i)$ and $\mathbb{Q}(e^{\frac{2\pi i}{3}})$ have class number one, we show they are Euclidean domains by tessellating the complex plane with the points $a+bv : a, b \in \mathbb{Z}$, where $1, v$ is…
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Where Fermat's last theorem fails

It's fairly well known that Fermat's last theorem fails in $\mathbb{Z}/p\mathbb{Z}$. Schur discovered this while he was trying to prove the conjecture on $\mathbb{N}$, and the proof is an application of one of his results in Ramsey theory, now known…
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Motivation for Eisenstein Criterion

I have been thinking about this for quite sometime. Eisentein Criterion for Irreducibility: Let $f$ be a primitive polynomial over a unique factorization domain $R$, say $$f(x)=a_0 + a_1x + a_2x^2 + \cdots + a_nx^n \;.$$ If $R$ has an irreducible…
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About the localization of a UFD

I was wondering, is the localization of a UFD also a UFD? How would one go about proving this? It seems like it would be kind of messy to prove if it is true. If it is not true, what about localizing at a prime? Or what if the UFD is Noetherian?
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Prove that a UFD is a PID if and only if every nonzero prime ideal is maximal

Prove that a UFD is a PID if and only if every nonzero prime ideal is maximal. The forward direction is standard, and the reverse direction is giving me trouble. In particular, I can prove that if every nonzero prime ideal is maximal then every…
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$A=\frac{\mathbb{C}[X,Y]}{(X^2+Y^2-1)}$ is a PID.

I was given an exercise to show $A=\frac{\mathbb{C}[X,Y]}{(X^2+Y^2-1)}$ is a PID. But I wonder if it is at all true. Note that PID $\implies$ UFD. But we have $$X\cdot X = 1-Y^2 =(1-Y)(1+Y)$$ in $A$ which contradicts UFD. Is there something wrong…
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Motivation behind the definition of GCD and LCM

According to me, I can find the GCD of two integers (say $a$ and $b$) by finding all the common factors of them, and then finding the maximum of all these common factors. This also justifies the terminology greatest common divisor. However, the…
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Why is the ring of holomorphic functions not a UFD?

Am I correct or not? I think that a ring of holomorphic functions in one variable is not a UFD, because there are holomorphic functions with an infinite number of $0$'s, and hence it will have an infinite number of irreducible factors! But I am…
Marso
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Irreducibles are prime in a UFD

Any irreducible element of a factorial ring $D$ is a prime element of $D$. Proof. Let $p$ be an arbitrary irreducible element of $ D$. Thus $ p$ is a non-unit. If $ ab \in (p)\smallsetminus\{0\}$, then $ ab = cp$ with $ c \in D$. We write $…
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Does UFD imply noetherian?

It is easy to show that a PID must be noetherian. My question is: Does UFD imply noetherian? If not, is there an easy counterexample? I apologize if this turns out to be a simple question. Thanks in advance!
minimax
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Is the coordinate ring of SL2 a UFD?

Is the ring $K[a,b,c,d]/(ad-bc-1)$ a unique factorization domain? I think this is a regular ring, so all of its localizations are UFDs by the Auslander–Buchsbaum theorem. However, I know there are Dedekind domains (which are regular; every local…
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Does $A$ a UFD imply that $A[T]$ is also a UFD?

I'm trying to prove that $A$ a UFD implies that $A[T]$ is a UFD. The only thing I am sure I could try to use is Gauss's lemma. Also, how can we deduce that the polynomial rings $\mathbb{Z}[x_1,\ldots,x_n]$ and $k[x_1,\ldots,x_n]$ are UFDs?
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An example of a non Noetherian UFD

An example of a non Noetherian UFD. I know an example is $$K[x_1,\ldots,x_n,\dots]$$ with $K$ a field, but I don't know why. Can someone give another example or better an explanation? Is it not Noetherian because it's not finitely generated? And…
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What is the correct notion of unique factorization in a ring?

I was recently writing some notes on basic commutative ring theory, and was trying to convince myself why it was a good idea to study integral domains when it comes to unique factorization. If $R$ is a commutative ring, and $a$ is a zero divisor, we…
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