# Tagged Questions

**2**

votes

**2**answers

149 views

### Domains $D$ for which for any prime $P$, $D_P$ is a PID

Is there any name or alternative characterization for the class of integral domains $D$ such that for any prime ideal $P$, $D_P$ is a principal ideal domain?

**3**

votes

**1**answer

349 views

### Prime ideals in the ring of algebraic integers

Let $m(x) = x^n + a_{n-1}x^{n-1} + \dots + a_1 x+ a_0$, $a_i \in \mathbb{Z}$, be an irreducible polynomial over $\mathbb{Q}$ and $K = \mathbb{Q}(x) / {m(x)\mathbb{Q}(x)}$. K is an algebraic number ...

**3**

votes

**1**answer

123 views

### How to estimate a local hilbert samuel funcion

Let $X$ be a reduced hypersurface in the projective variety $\mathbb{P}^n(K)$, where $K$ is a number field. Select $\xi$ is a $F_{\mathfrak{p}}$-rational point of $X$ where $\mathfrak{p}$ is a prime ...

**14**

votes

**1**answer

710 views

### Principal maximal ideals in Z[x]/(F)

Is there some irreducible $F \in \mathbb{Z}[x]$ such that $\mathbb{Z}[x]/(F)$ has no principal maximal ideal? Equivalently, is it possible that the $1$-dimensional integral domain $\mathbb{Z}[x]/(F)$ ...

**4**

votes

**3**answers

408 views

### Orders of Number Fields

Let $K$ be a number field over $\mathbb{Q}$ of degree $n$, and $\mathcal{O} \subset \mathcal{O}_K$ an order.
$\textbf{Questions:}$
$\newcommand{\Spec}{\textrm{Spec }}$
$\newcommand{\cO}{\mathcal{O}}$
...

**16**

votes

**3**answers

1k views

### Commutative Algebra with a View Toward Algebraic Number Theory

Someone asked me this today, and I don't know what the standard answer is:
Is there an analogue of David Eisenbud's rather amazing Commutative Algebra With a View Toward Algebraic Geometry but with a ...

**6**

votes

**2**answers

715 views

### Decomposition of finite algebras over finite fields

Let $K$ be a number field, $Z_K$ its ring of integers, and $p$ a rational prime number. Then $A_p = Z_K/(p)$ is a finite ${\mathbb F}_p$-algebra. Using ideal arithmetic in $Z_K$ and the Chinese ...

**4**

votes

**0**answers

267 views

### What to call the following variant of tame ramification

Suppose that $R \subseteq S$ is a generically separable extension of 1-dimensional normal domains (you can assume that $R$ is local if you'd like) of equal-characteristic $p > 0$ (for simplicity, ...

**6**

votes

**1**answer

384 views

### The Jacobian ideal generates the socle of a complete intersection

This is with reference to theorem 5.20 in Vasconcelos book linked (google books) here:
http://tinyurl.com/2967eov
I shall restate the theorem here for easy reference: "If $A=k[[x_1,x_2,...,x_n]]/I$ ...

**4**

votes

**3**answers

370 views

### Examples of DVRs of residue char p and ramification e

I am looking for concrete examples of a complete discrete valuation ring $R$ of characteristic 0, residue characteristic $p$ and ramification index $e$. By residue characteristic, I mean the ...

**6**

votes

**2**answers

330 views

### Does regularity of a prime ideal in the fibre imply regularity of the prime?

Recall that a prime $\mathfrak{p}$ is called nonsingular (regular) if the localization at that prime is a regular local ring. If all primes of a ring $R$ are nonsingular, $R$ is called regular. Let ...

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vote

**0**answers

140 views

### Ideals weak equivalence and “finite” equivalence

Let $R$ be an order in a number field. Two $R$-ideals $I$ and $J$ are weak equivalent if there exist (necessarily invertible) ideals $X$ and $Y$ such that $I X=J$ and $J Y=I$.
This is equivalent to ...

**8**

votes

**4**answers

1k views

### Is there much difference between Kronecker's and Dedekind's methods in algebraic number theory and commutative algebra?

Edwards, in his book "Divisor theory" says that Kronecker's methods are quite different to Dedekind's and those of today. Is there really much of a difference apart from Kronecker's methods being more ...

**3**

votes

**1**answer

358 views

### Non-existence ofintegral basis of integral closure in a finite extension of Frac(A), A Dedekind.

Let $A$ be a Dedekind domain, $K:=\text{Frac}(A)$ and $L/K$ finite so that the integral closure $B$ of $A$ in $L$ is Dedekind. If $A$ is a PID, for example, then there exists an integral basis : $B$ ...

**9**

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**1**answer

419 views

### How to find examples of non-trival kernel of maps between Brauer groups Br(R) -> Br(K)

Background/Motivation: The facts about the Brauer groups I will be using are mainly in Chapter IV of Milne's book on Etale cohomology (unfortunately it was not in his online note).
Let $R$ be a ...