Let $R$ denote a UFD, and let $X = \{x_0,\cdots,x_{n-1}\}$ denote a finite set of symbols. Then $R[X]$ is a UFD. This follows, since if $R$ is a UFD, then so too is $R[x],$ for any symbol $x \notin R$. Q. If $R$ is a UFD, and $X$ is an arbitrary set, possibly infinite, is […]

This question already has an answer here: Prove that any polynomial in $F[x]$ can be written in a unique manner as a product of irreducible polynomials in F[x]. 1 answer

This question already has an answer here: Prove that $k[x,y,z,w]/(xy-zw)$, the coordinate ring of $V(xy-zw) \subset \mathbb{A}^4$, is not a unique factorization domain 1 answer

Let $D$ be a UFD, let $F$ be the field of fractions of $D$, let $a \in D$ be such that $x^2 \ne a, \forall x \in D$. Then is it true that $x^2\ne a ,\forall x \in F$ ? (This problem is motivated from the fact that if square root of an integer is […]

One of the defining features of a UFD is that any height one prime ideal is principal (see Wikipedia). Is it also true that any height one (i.e. every prime minimal among those containing it has height $1$) ideal is principal? Is this true even in the case of polynomial rings? If not, please provide […]

Prove that the quotient ring $\mathbb{C}[x,y]/(x^2+y^2-1)$ is a unique factorization domain. I am trying to prove first it is a principal ideal domain. However I am really stuck on this problem

I’m asked to show that 3 is irreducible but not prime in $R = \mathbb{Z}[\sqrt{-41}]$. And if $R$ is a Euclidean domain. To show that it’s not prime I have $(1 + \sqrt{-41})(1 – \sqrt{-41}) = 42 = (3)(14)$. I get that 3 divides 14 but how do I show that 3 does not divide […]

I read that Grothendieck developed Local Cohomology to answer a question of Pierre Samuel about when certain type of rings are UFDs. I know the basics of local cohomology but I have not seen a theorem which shows the connection between UFDs and Local Cohomology. My Question: Could someone tell me about a result which […]

The proof (at least the proof I know) that a principal ideal domain is a unique factorization domain uses the axiom of choice in multiple ways, and the usual way to show that a Euclidean domain is a UFD is to show that it’s a PID (which is easy and constructive). Is there a direct […]

As the title says, I’m trying to prove that $k[[X_1,\ldots,X_n]]$ is UFD for $k$ field. In Lang’s Algebra, there is a proof by induction on $n$. The base of induction is clear (we even have discrete valuation ring). Let $R_n = k[[X_1,\ldots,X_n]] \cong R_{n-1}[[X_n]]$ and assume that $R_{n-1}$ is UFD. Let $f\in R_n$, $f(X_1,\ldots,X_n)\neq 0$. […]

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