This is a part of statement in the book I do not fully appreciate. Suppose $E/K$ is Galois extension and $G$ the Galois group of $E/K$. $E[G]$ is the group ring formed by finite group $G$. "It is worthwhile remarking that $E\otimes_KE\cong EG$ can be viewed as a deep reason why Galois theory wor...

All rings in this post are commutative and with $1$. Everyone knows the definition of a factorial ring, a. k. a. unique factorization domain (UFD). I have been wondering about some variations regarding this notion. (a) A ring $R$ is called pre-pre-Schreier (this is my nomenclature) if and only ...

Suppose that $X=Spec(A)$ is an affine variety over an algebraically closed field $k$ which is normal and such that $Cl(X)=0$. I am interested in hypersurfaces of $X$ which again satisfy this condition. Q: Do there always exist hypersurfaces $X'\subset X$ such that $X'$ is again normal and $Cl...

The following result is too elementary, both to state and to prove, not to be known. Can someone give a reference? Is there any hope if you don't suppose UFD (i.e. move that from the hypothesis to the conclusion)? Theorem. Let $R$ be a commutative UFD with field of fractions $F$. Suppose tha...

Lemma 3.2 says: Let $A$ be a UFD. Let $R \subseteq A$ be a subring of $A$ such that $R^* = A^*$. The following conditions are equivalent: (i) Every irreducible element of $R$ remains irreducible in $A$. (ii) $R$ is factorially closed in $A$, namely, if $x,y \in A$ satisfy $xy \in R - {0}$, then...

Let $w$ be an algebraic element over $\mathbb{C}[x]$, with minimal polynomial $f(t)=c_mt^m+\cdots+c_1t+c_0$, $c_i \in \mathbb{C}[x]$. Is it possible to characterize (in terms of the $c_j$'s) all algebraic elements $w$, such that $R=\mathbb{C}[x][w]$ has no prime elements? Please see this re...

Let $f=f(T) \in \mathbb{C}[x,y][T]$ be a monic irreducible polynomial: $f=T^n+a_{n-1}T^{n-1}+\cdots+a_1T+a_0$, $a_j \in \mathbb{C}[x,y]$, $0 \leq j \leq n-1$. Denote $B=\mathbb{C}[x,y,T]/(f)=\mathbb{C}[x,y][w]$, where $w^n+a_{n-1}w^{n-1}+\cdots+a_1w+a_0=0$. Of course, $\mathbb{C}[x,y]$ is a UFD...

Let $A \subseteq A[w] \subseteq C$ be three Noetherian integral domains, over a field $k$ of characteristic zero, with $A \subseteq C$ an algebraic ring extension (in particular, $w$ algebraic over $A$). Further assume that: (1) $A$ and $C$ are unique factorization domains (UFD's). (2) $A \subs...

Looking through some old notes of mine from two years ago I found some crude notes writing what amounted to the statement that for any prime $p\equiv 1\bmod 4$ one could express for any odd integer $p\nmid n$ both of the order four dirichlet characters $\chi_1,\chi_2:(\mathbb{Z}/p\mathbb{Z})^{*}\...

7 is a Heegner number. Therefore the integer ring $O_K$ corresponding to $K=\mathbb{Q}[\sqrt{-7}]$ is a unique factorization domain. Now, it is easy to show that $\mathbb{Z}[\sqrt{-7}]\subset O_K$, i.e. $O_K \neq \mathbb{Z}$, because $x=\sqrt{-7}$ is a solution to $x^2+7=0$. On the other hand,...

$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$ shows that it is possible to have commutative ring $R$ with unity such that $R \cong R[X,Y]$ but $R \ncong R[X]$. My questions are: Is it possible to have an example of a commutative ring with unity $R$ such that $R \cong R[X,Y]$ but $R \ncong R...

Everybody knows that polynomial rings over fields have unique factorization, and that if $R$ has unique factorization, then so does $R[X]$. And everybody knows who proved these results first. Well, perhaps not. Which brings me to my questions: Who first proved that $k[x]$ is factorial if $k$ ...

In Peter Bundschuh's "Einführung in die Zahlentheorie" I came across a possibly well-known but to me rather peculiar proof of unique factorisation, which is attributed to Ernst Zermelo. The proof bypasses Euclid's lemma to prove that $ \mathbb{Z}$ is a UFD. It seems to me that the deepest propert...

Let $R$ be a commutative ring with identity. Then $R$ is $\textit{catenary}$ if for each pair of prime ideal $p \subsetneq q$, all maximal chains of prime ideals $p = p_0 \subsetneq p_1 \subsetneq \dots \subsetneq p_n = q$ have the same length. In some (informal) texts the author conclude that ...

Let $c$ be an integer, not necessarily positive and $|c|$ not a square. Let $\mathbb{Z}[\sqrt{c}]$ be the set of complex numbers $$a+b\sqrt{c}, \quad a, b\in \mathbb{Z},$$ which form a subring of the ring $\mathbb{C}$ under the usual addition and multiplication. Are the following questions comp...

The question is in the title, but employs some private terminology, so I had better explain. Let $R$ be an integral domain with fraction field $K$, and write $R^{\bullet}$ for $R \setminus \{0\}$. For my purposes here, a norm on $R$ will be a function $| \ |: R^{\bullet} \rightarrow \mathbb{Z}^...

Can anyone suggest a good place to read up on the number theoretic properties of and techniques for $\mathbb{Z}[1/p]$, (that is, rational numbers with only powers of a prime $p$ in the denominator)? I find myself struggling to answer some of the more basic questions about this ring, especially w...

An integral domain $R$ is said to be Euclidean if it admits some Euclidean norm: i.e., a function $N: R \rightarrow \mathbb{N} = \mathbb{Z}^{\geq 0}$ such that: for all $x, y \in R$ with $N(y) > 0$, either $y$ divides $x$ or there exists $q \in R$ such that $N(x-qy) < N(y)$. A well-known "descen...

Let $n$, $k$ and $m_1, \dots, m_k$ be positive integers. Which is the most efficient algorithm to find out whether there are positive integers $a_1, \dots, a_k$ such that $n = \sum_{i=1}^k a_i m_i$? To make things nontrivial, think of $k$ being in the hundreds, and of $n$ and the $m_i$ having hu...

I have posted this question at stackexchange (502413), without responses until now. In the papers by T. Motzkin: The Euclidean Algorithm, Bull. AMS 55, 1949, pp. 1142--1146 and P. Samuel: About Euclidean Rings, Journ. of Alg. 19, 1971, pp. 282--301, it is shown that among all the Euclidean algor...

I asked a similar question a few weeks ago in M.SE but it didn't receive any answers, so I decided to post it here with some modifications. My motivation comes from a theorem given in Pete L. Clark's notes on factorization. More exactly, I'm referring to the theorem 46 and it says this: for a Bé...

It is well known that given two nonconstant polynomials $f,g\in F[x]$ where $F$ is a field, there are unique polynomials $r_0,\dots ,r_n$ such that $$f=r_n g^n +\dots+r_1 g +r_0,$$ where $\deg r_i <\deg g$ for all $i$. In other words, the radix expansion is possible in the ring $F[x]$. The pro...

It is known that there are finitely many square-free integers $d$ for which the ring of integers of $\mathbf{Q}(\sqrt{d})$ is Euclidean under the norm function. Is there an analogous result for quadratic extensions of $K(t)$, $t$ an indeterminate and $K$ a (finite) field? Also, can we c...

Twin primes, like (29, 31) and (137, 139) are interesting to study. I have been exploring the parallels of the Gaussian and Eisenstein integers with the rational integers. For instance, they have primes and composites in common. But are there "twin primes"? There may not be a direct analogy, sinc...

I am encountering a geometrical question that intuitively seems obvious but I have a lack of argument to prove or disprove it in a rigorous manner. Let $\Omega\subset\Bbb R^d$ be an open bounded (may be connected just to make it simpler) For $\delta>0$ small enough we define the shrunken versi...

Let $\Omega\subset \Bbb R^d$ be a bounded $C^1$ domain. Let $u:\Bbb R^d\to \Bbb R$ be a function in $C^2_b(\Bbb R^d)$. I would like to compute the following limit: for $x\in \partial \Omega$ $$L= \lim_{s\to 1}(1-s)\int_{\Omega}\frac{(u(x)-u(y))}{|x-y|^{d+2s}} d y. $$ Here is what I did so far:...

Let $c$ be an integer, not necessarily positive and $|c|$ not a square. Let $\mathbb{Z}[\sqrt{c}]$ be the set of complex numbers $$a+b\sqrt{c}, \quad a, b\in \mathbb{Z},$$ which form a subring of the ring $\mathbb{C}$ under the usual addition and multiplication. Are the following questions comp...