Tagged Questions

3
votes
1answer
48 views

Irreducible ideal implies prime ideal in Dedekind Domains?

An ideal is irreducible if it can not be written as the finite intersection of strictly larger ideals. In a Noetherian ring every irreducible ideal is primary, but the converse doesn't hold. I wonder ...
7
votes
2answers
67 views

Show field of fractions is finite extension of $\mathbb{Q}$

Let $A$ be a ring which is also a finitely generated $\mathbb{Z}$-module. If $A$ is an integral domain and $K$ is its field of fractions and $K$ has characteristic zero, then why is $K$ a finite ...
5
votes
2answers
72 views

Algorithmic approach to enumerating ideals in $\Bbb Z[x]/(m, f(x))$

I'm studying for my algebra quals this fall and keep encountering problems like the following: List all the ideals of $\mathbb{Z}[x]/(16, x^3)$. or List all the primes of ...
1
vote
1answer
31 views

If $R$ is integral over $S$, then $frac(R)/frac(S)$ is finite extension of fields

How to show that: If $R\supset S$ are rings, $R$ is integral over $S$, $K$ and $L$ the fraction fields of $R$ and $S$ respectively, then $K/L$ is finite extension of fields.
3
votes
2answers
51 views

Is it true that $\mathbb{Z}_{(p)}=\mathbb{Z}_{p}\cap \mathbb{Q}$?

I know $\mathbb{Z}_{(p)}\subset \mathbb{Z}_{p}\cap \mathbb{Q}$, where $\mathbb{Z}_{(p)}$ is the localization of $\mathbb{Z}$ at prime ideal $(p)$ and $\mathbb{Z}_p$ is the set of p-adic integers. I ...
5
votes
2answers
105 views

Ideals in a Dedekind domain localized at a prime ideal

Let $R$ be a Dedeking domain, let $\mathfrak{i}$ be a non-zero ideal of $R$. By factorization theorem we can write $$\mathfrak{i}=\mathfrak{p}_1^{a_1}\cdots\mathfrak{p}_n^{a_n}$$ for distinct non-zero ...
2
votes
1answer
47 views

A necessary and sufficient condition for a full lattice over an integral domain

I'm learning about lattices over integral domains and I would be grateful if someone could clarify the following for me. Let $R$ be an integral domain with quotient field $K$ where $K\neq R$. Suppose ...
5
votes
1answer
79 views

$P/P^2$ isomorphic to $R/P$ as $R$-modules

Let $P$ be an ideal of a ring $R$. When is it true that $P^n/P^{n+1}$ are isomorphic to $R/P$ as $R$-modules for any $n$? I was trying to show that for Dedekind domains the norm of ideals is a ...
5
votes
2answers
109 views

Integral domains such that all proper factor rings are finite

Let $\mathbb Z$ be the ring of rational integers. If $a\in\mathbb Z$ is a non-zero element, then the factor ring $\mathbb Z/(a)$ is finite and has order $|a|$. If $\mathbb Z[i]$ is the ring of ...
4
votes
2answers
63 views

Question regarding two equivalent definitions of Dedekind domains

Theorem: If $A$ is a Noetherian integral domain, the following two properties are equivalent. 1) $A_{\mathfrak p}$ is a DVR for every prime ideal $\mathfrak p \neq 0$; 2) $A$ is ...
3
votes
1answer
143 views

Multidimensional Hensel lifting

I have a question about a practical application of (some) generalised form of Hensel's Lemma. I cannot find it stated in an appropriate form in Bourbaki or anywhere else, so here goes ... Let $p$ be ...
-3
votes
1answer
138 views

Show that $p$ and $q$ are not principal, but that $p^2$, $pq$ and $q^2$ are.

Let $K$ be the field $\mathbb Q(\sqrt{−15})$, let $R = \mathcal{O}_K$ be the ring of integers of $K$. Let $\alpha= \frac{-1+\sqrt{-15}}{2}$ and consider the prime ideals $p = (2,α)$ and $q = (17,α + ...
1
vote
3answers
132 views

Showing an ideal is a projective module via a split exact sequence

Let $R=\mathbb{Z}[\sqrt{-6}]$ and $I=(2,\sqrt{-6})$ the ideal generated by 2 and $\sqrt{-6}$. I want to show that $I$ is not a projective $R$-module by producing a short exact sequence that splits, ...
0
votes
0answers
44 views

Show that $\mathcal{O}^+_K$ contains $\mathcal{O}_K$, and that the discriminant $\Delta(K)$ is the index $[\mathcal{O}^+_K : O_K]$. [duplicate]

Let $K$ be a number field, let $\mathcal{O}_K$ be its ring of integers, and let $B = \{b_1,\ldots,b_d\}$ be a subset of $K$ of cardinality $d$ such that $\mathcal{O}_K = ...
4
votes
3answers
214 views

Does any integral domain contain an irreducible element?

Let $R$ be an integral domain which is not a field. Does $R$ necessarily have an irreducible element? I suspect the answer is no, but I couldn't find an example showing that...
-2
votes
1answer
97 views

Prime ideal in a Dedekind domain

Let $\mathfrak p $ be a prime ideal in a Dedekind domain $O$ with field of fractions $K$. Define $$\mathfrak p^{-1}= \{x \in K: x\mathfrak p \subset O\}.$$ Let $\mathfrak a \subset \mathfrak p$ and ...
2
votes
2answers
190 views

Book recommendations for commutative algebra and algebraic number theory

Are there any books which teach commutative algebra and algebraic number theory at the same time. Many commutative algebra books contain few chapters on algebraic number theory at end. But I don't ...
3
votes
1answer
83 views

$I|J \iff I \supseteq J$ using localisation?

Let $R$ be a Dedekind domain. We know that for ideals $I$ and $J$ of $R$ that $I|J \iff I \supseteq J$. This fact is used for example in Marcus' Number Fields to show that we have unique factorisation ...
0
votes
1answer
31 views

If ideals $Q_1,Q_2$ lie over a prime in $\Bbb{Z}$ their product lies over the prime squared?

Suppose we have a Dedekind domain $R$ which for the moment we can take to be $\mathcal{O}_K$ for some algebraic number field $K$. Now suppose that $Q_1,Q_2$ are prime ideals that lie over a prime ...
2
votes
2answers
71 views

Can a ring of integers contain a $2$-dimensional noetherian normal integral domain?

Let $K$ be a number field with ring of integers $O_K$. Does there exist a $2$-dimensional subring $A\subset O_K$? Clearly, if such a subring $A\subset O_K$ exists, we have that $A$ is an integral ...
2
votes
3answers
107 views

Atiyah - Macdonald Exericse 9.7 via Localization

I am trying to show that the quotient of a Dedekind domain $A$ by an ideal $\mathfrak{a}$ is a PIR (principal ideal ring). Now by using the Chinese Remainder Theorem and the fact that a direct product ...
3
votes
1answer
97 views

Exercise from Matsumura about DVRs

Another result I would really appreciate some help with: Suppose $R$ is a DVR and let $K$ be its field of fractions. Let $L$ be a finite extension of $L$. Prove that any valuation domain inside of ...
1
vote
2answers
114 views

$xy\in (x^2,y^2)$ if $R$ is a Dedekind domain

I would really like to see a simple proof for the following question, if possible. Let $R$ be a Dedekind domain. Then, $xy \in (x^2,y^2)R$ for any $x,y$ in $R$. Also, show that this fails in general ...
2
votes
1answer
72 views

Integral basis of an extension of number fields

Let $K\subseteq F$ be number fields with ring of integers $\mathcal{O}_K\le \mathcal{O}_F$. Question: Is $\mathcal{O}_F$ a free $\mathcal{O}_K$-module ? By the integral basis theorem this is true ...
4
votes
1answer
62 views

Is the ring of all algebraic integers coherent?

Is the ring of all algebraic integers coherent? Here is the definition of a coherent ring.
1
vote
2answers
81 views

Why are distinct maximal ideals coprime?

I have a problem. As I've understand it two proper ideals $I$ and $J$ of a ring $R$ is said to be $coprime$ if $I+J=(1)$. For a set of distinct maximal ideals of $R$ say $\{I_i\}$, $0\leq i\leq n$, ...
3
votes
2answers
221 views

How can I prove that every maximal ideal of $B= \mathbb{Z} [(1+\sqrt{5})/2] $ is a principal?

How can I prove that every maximal ideal of $B= \mathbb{Z} [(1+\sqrt{5})/2] $ is a principal? I know if I show that B has division with remainder, that means it is a Euclidean domain. It follows that ...
2
votes
1answer
92 views

How to make sense of the ideal norm in a localization?

This is a follow-up to this question: Localization of finite modules, or: compatibility of ideal norms with localization at a prime number Let $K$ be an algebraic number field and ...
2
votes
1answer
191 views

A non-degenerate trace implies dual basis [updated]

I found a better proof of the theorem in Serge Lang - Algebraic Number Theory but I put in bold the parts I don't understand. Hoping for any explanations of these points. The trace $Tr : L \to K$ ...
4
votes
1answer
52 views

Example of an excellent Henselian regular local ring containing a field that is not the formal power series ring

I am reading this paper http://arxiv.org/pdf/0709.3628.pdf and I was trying to construct examples of excellent Henselian regular local rings containing a field that are not complete, but could not ...
4
votes
2answers
174 views

Galois theorem for ideals?

Let $R$ be an integrally closed integral domain with fraction field $K$. Let $L$ be a finite Galois extension of $K$ and let $\sigma_1,\dots,\sigma_n$ be the elements of $Gal(L/K)$. Let $S$ be $R$'s ...
1
vote
1answer
61 views

Extension of rings of integers always locally free

In his answer to this question, Andrea claims that if $A \subset B$ is an extension of rings of integers of number fields, $B$ is locally free over $A$. How can one prove this? Furthermore, I am ...
2
votes
1answer
91 views

If $S$ is integral over integrally closed $R$, can $\mathfrak{a}S$ be principal without $\mathfrak{a}\triangleleft R$ being principal?

Let $R\subset S$ be integral domains, with $R$ integrally closed in its field of fractions, and $S$ integral over $R$. Suppose that the fraction field of $S$ is a finite Galois extension of the ...
7
votes
2answers
316 views

Classgroup of $\mathbb{Q}(\sqrt{2},\sqrt{-13})$

How would you compute the classgroup of the biquadratic number field $\mathbb{Q}(\sqrt{2},\sqrt{-13})$? I would prefer a method as "from scratch" as possible. Please avoid, if possible, quoting ...
4
votes
1answer
126 views

The ring of integers of a number field is finitely generated.

For a number field $K$, we define the ring of integers of $K$ to be $$\mathcal{O}_K:=\{x\in K\big|\ (\exists f\in\mathbb{Z}[X])(f\ \text{ is monic and } f(x)=0)\}.$$ Is there any easy way to see from ...
1
vote
2answers
157 views

Discrete valuation ring associated with a prime ideal of a Dedekind domain

Let $A$ be a Dedekind domain. Let $K$ be the field of fractions of $A$. Let $P$ be a non-zero prime ideal of $A$. Let $v_P$ be the valuation of $K$ with respect to $P$. Then the localization $A_P$ of ...
5
votes
1answer
138 views

Two corollaries in Lang's Algebraic Number Theory.

I'm having difficulty understanding the relationship between two corollaries in Lang's Algebraic Number Theory, on page 16 for those with the book. They can also be found in his Algebra. The first ...
2
votes
1answer
84 views

Technical question on integral ring extensions

Let $A$ be an integral domain, integrally closed in its field of quotients $K$ and let $L$ be a finite Galois extension of $K$ with group $G$. Let $B$ be the integral closure of $A$ in $L$. Let $p$ be ...
5
votes
2answers
197 views

Question about $p$-adic numbers and $p$-adic integers

I've been trying to understand what $p$-adic numbers and $p$-adic integers are today. Can you tell me if I have it right? Thanks. Let $p$ be a prime. Then we define the ring of $p$-adic integers to ...
5
votes
3answers
303 views

On the ring generated by an algebraic integer over the ring of rational integers

Let $f(X) \in \mathbb{Z}[X]$ be a monic irreducible polynomial. Let $\theta$ be a root of $f(X)$. Let $A = \mathbb{Z}[\theta]$. Let $p$ be a prime number. Suppose $p$ does not divide the discriminant ...
1
vote
0answers
70 views

Generalized Hensel lifting in a Dedekind domain

The following theorem is known as generalized Hensel lifting(see here). Can we prove this without using $P$-adic completion? Theorem Let $A$ be a Dedekind domain. Let $P$ be a non-zero prime ideal of ...
1
vote
0answers
67 views

Two questions concerning integral dependence

Proposition 2.4 in Janusz's Algebraic Number Fields states that if $R$ is an integral domain with quotient field $K$, $L/K$ a field extension and $b \in L$ algebraic over $K$ with minimal polynomial ...
1
vote
1answer
209 views

An exact sequence on the ideal class group of a Noetherian domain of dimension 1

Let $A$ be a Noetherian domain of dimension 1. Let $K$ be its field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is finitely generated as an $A$-module. It is well-known ...
4
votes
1answer
105 views

Embedding torsion units of an order into torsion units of the reduced order.

Let $A$ be an order, i.e. a commutative ring of which the additive group is isomorphic to $\mathbb{Z}^n$ for a certain non-negative integer $n$. Show that there exists an embedding ...
19
votes
1answer
346 views

Connectedness of the spectrum of a tensor product.

Let $A$, $B$ be finite, free $\mathbb{Z}$ algebras such that $\operatorname{Spec}(A)$ and $\operatorname{Spec}(B)$ are both connected. Is $\operatorname{Spec}(A\otimes_{\mathbb{Z}} B)$ connected?
1
vote
1answer
209 views

A lemma on the integral closure of a Noetherian domain of dimension 1

I need to prove the following lemma(?) which is motivated by this and this. Lemma Let $A$ be a Noetherian domain of dimension 1. Let $K$ be the field of fractions of $A$. Let $B$ be the integral ...
10
votes
2answers
601 views

Ideal class group of a one-dimensional Noetherian domain

Let $A$ be a one-dimensional Noetherian domain. Let $K$ be its field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is finitely generated $A$-module. It is well-known that B ...
2
votes
2answers
120 views

Is the ring of integers in a relative algebraic number field faithfully flat over a ground ring?

Let $L$ be a finite extension of an algebraic number field $K$. Let $A$ and $B$ be the rings of integers in $K$ and $L$ respectively. Is $B$ faithfully flat over $A$? What if $L$ is an infinite ...
5
votes
3answers
365 views

Prime ideals in the ring of algebraic integers

Let $\mathcal{O}$ be the ring of all algebraic integers: elements of $\mathbb{C}$ which occur as zeros of monic polynomials with coefficients in $\mathbb{Z}$. It is known that $\mathcal{O}$ is a ...
2
votes
0answers
100 views

A yet another theorem on the different ideal of algebraic number fields

I think I came up with a proof of the following theorem using non-archimedian completions. But I'm not 100% sure. Is this correct? Theorem Let $A$ be a Dedekind domain, $K$ its field of fractions. ...

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