3
votes
2answers
54 views

Does it hold that the $p$-adic completion of the integers equals the completion of the localization in $p$?

maybe this is a stupid question, but I could not solve it even for the ordinary integers $\mathbb{Z}$. Furthermore, I don't have to much knowledge on algebraic number theory and ramifications. Let ...
2
votes
1answer
54 views

Strong approximation theorem for Dedekind Domains

This is a theorem in "Maximal Orders" by Reiner. Page 48 stated without proof. And is said to be an easy consequence of The Chinese remainder Theorem. I am attempting to prove the theorem and need a ...
3
votes
0answers
19 views

Do lattices in a field of fractions contain an ideal?

Let $R$ be a noetherian commutative integrally closed domain whose field of fractions $K$ is a finite extension of the field of fractions $Q$ of $\Lambda = \mathbb{Z}_p[[T]]$. Let $L \subset R$ be a ...
3
votes
1answer
75 views

How to prove this innocent looking isomorphism

I've got a Dedekind domain $R$ with quotient field $K$, a non-zero prime ideal $P$ of $R$. I form the completion $\widehat{K}$ of $K$ wrt the valuation $v_P$ associated to $P$. Let $\widehat{R}$ be ...
4
votes
3answers
127 views

Is $\mathbb{Z}[\sqrt{2} + \sqrt{3}]$ integrally closed?

Is $\mathbb{Z}[\sqrt{2} + \sqrt{3}]$ integrally closed? (Or could it have a relation to another domain like $\mathbb{Z}[\sqrt{-3}]$ does with $\mathbb{Z}[\omega]$?) Also, is it UFD? What are its ...
3
votes
1answer
41 views

Rings of algebraic integers

A basic question on algebraic numbers. If $L/K$ is a finite extension of number fields with respective rings of integers $\mathcal O_L$ and $\mathcal O_K$ then is it true that $\mathcal O_L$ is ...
4
votes
1answer
104 views

Localization in formal power series

I saw in a textbook the following assertion: Let $R$ be a commutative ring with unity, and $R[[X]]$ be the ring of power series in one indeterminate $X$. If the homomorphism $\phi‚ą∂ R[[X]] \to R$ ...
7
votes
2answers
184 views

Why study integrality?

Here are a few of the basic definitions related to integrality. (1) A polynomial in $R[x]$ is monic if its leading coefficient is $1$. (2) An element is integral over a ring $R$ if it ...
1
vote
1answer
48 views

Degree of extension is equal to linear combination of prime factor multiplicities with prime factor index coefficients in Dedekind domains

I'm working on the following problem... Suppose that $A$ is a Dedekind domain with fraction field $K$. $L/K$ is a finite separable extension of $A$ of degree $n$, and $B$ is the integral closure ...
0
votes
1answer
71 views

$(p)$ is a prime ideal in $\mathbb{Z}[\sqrt{n}]$ if and only if $n$ is not a square mod $p$

Let $n$ be a square-free integer such that $n\equiv 2,$ or $3\bmod 4$. If $p\in\mathbb{Z}$ be a prime such that $n$ is not a square modulo $p$, then $(p)$ is a prime ideal in ...
1
vote
1answer
92 views

When does coprimality carry over to the base ring in an integral extension of Dedekind domains?

Let $A$ be a Dedekind domain. Let $K$ be the field of fractions of $A$ and $L$ is some finite field extension of $K$. Then let $B$ be the integral closure of $A$ in $L$. (Sorry I don't know how to ...
2
votes
1answer
66 views

Exact sequence out of commutative exact diagram

I'm trying to get grip on the following commutative exact diagram: I know where the maps come from and could verify the exactness and the other maps. (It is induced by the long exact sequence of ...
2
votes
2answers
166 views

A quotient $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain is principal (Neukirch exer 1.3.5)

The exercise states: The quotient ring $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain by an ideal $\mathfrak{a}\ne 0$ is a principal ideal domain. The proof by localization ...
2
votes
0answers
70 views

maximal vs non-maximal order in an algebraic number field

I am trying to determine whether an order in a (cubic) number field is maximal or not. I have picked up two different fields. One has a power basis the other does not have it. 1) Let ...
2
votes
1answer
87 views

Define, $p^{-1} = \{x \in K: xp \subset D\}$. Then show that there exists a non zero $c \in D$ such that $cp^{-1} \subset D$.

Let $D$ be an integral domain and $K$ be its field of fraction. Also, given that $D$ is Notherian, Integrally closed, and every non-zero prime ideal in $D$ is maximal ideal. Let $p$ be a ideal of ...
3
votes
1answer
62 views

Example 4.3.19 in Liu: unramification with schemes and numbers

In exemple 4.3.19 of Liu's book one hase $L/K$ an extension of number fields with integer rings $\mathcal{O}_L$ and $\mathcal{O}_K$, $\mathfrak{q}\subseteq\mathcal{O}_L$ a prime ideal and ...
2
votes
2answers
114 views

$p$-adic valuation.

Let $\alpha_1,\alpha_2\in \mathbb Z_p$ such that $v_p(\alpha_1)<v_p(\alpha_2).$ How to prouve that $v_p(\alpha_2-\alpha_1)=v_p(\alpha_1)$ ? I think this is a stupid question but I'm really ...
1
vote
1answer
83 views

Being Galois stable under completion?

Let $R$ a Dedekind Domain, $K = \mathrm{Frac}(R)$ the fraction field, $L/K$ a finite galois extension, $R'$ the integral closure of $R$ in $L$. Then $R'$ is Dedekind again. Let $\mathfrak{p} \subset ...
2
votes
0answers
30 views

$M\cong N$ iff $[M:N]_R$ is a principal fractional ideal

Let $R$ be a Dedekind ring, $K$ its field of fractions, $U$ a finite vector space over $K$, and $M,N$ finitely generated $R$-modules that span $U$, i.e. contain a basis of $U$. For every $\mathfrak p ...
1
vote
1answer
78 views

Questions about a proof in Greenberg's Book.

I am trying to understand the proof of the following lemma : Lemma ' : Suppose that $X$ is a finitely generated $\Lambda$-module ($\Lambda =\mathbb Z_p[[T]]$) and that ...
1
vote
1answer
82 views

The order of the cokernel of an endomorphism over $ \mathbb Z_p$

I want to prove the following result : Let $X$ a finite-rank free $\mathbb{Z}_p$-module, and $\varphi \colon X \to X$ an endomorphism of $X$. Then $$|M/\varphi(X)| < \infty \Leftrightarrow ...
0
votes
0answers
63 views

Conditions on $a,b\in\mathbb{Q}$, for $a+b\sqrt{n}$ to be integral over $\mathbb{Z}$

For $n\in \mathbb{Z}$ square-free, let $$k:=\mathbb{Q}(\sqrt{n}),$$ and let $$\alpha:=a+b \sqrt{n}\in k.$$ Prove that $$ \alpha \mbox{ is integral over } \mathbb{Z}\;\;\; \Longleftrightarrow ...
1
vote
1answer
103 views

Pro$-p-$group as a $\Lambda-$module

Let $p$ be a prime number, and let $X$ be an abelian pro$-p-$group (i.e for some indexing set $I,$ we have $X=\varprojlim X_i$ where $X_i$ is a finite, abelian $p-$group for each index $i \in I .$) ...
5
votes
2answers
128 views

$||x||=1$ in $K/\mathbb{Q}$ implies $x$ is a root of unity.

Let $K/\mathbb{Q}$ a finite (i.e. algebraic and finitely generated) extension. Let $x \in K$, such that $||x||=1$ for all normalized absolute values of $K$ but at most one. Then $x$ is a root of ...
0
votes
1answer
93 views

Power series ring over a ring of integers

Let $K/\mathbb {Q}_p$ be a finite extension, $\mathcal{O} := \mathcal{O}_K$ the ring of integers of $K,$ $\frak p$ the maximal ideal of $\mathcal{O}$, and $\pi$ a uniformizer, i.e., $\frak{p} = ...
4
votes
1answer
233 views

In a Dedekind domain every ideal is either principal or generated by two elements.

Prove that in a Dedekind domain every ideal is either principal or generated by two elements. Help me some hints. Thanks a lot.
2
votes
0answers
34 views

Why does taking the residue commute with the discriminant if $B$ is free over $A$ and not in general?

Let $K$ be a number field, $A$ its ring of integers, $L/K$ be a finite extension, and $B$ the integral closure of $A$ in $L$. Lemma (residue of the discriminant): Assume $B$ is free over $A$, let $a$ ...
4
votes
1answer
110 views

What is an example of a Dedekind ring that is non-principal?

Prop. 15 of Serge Lang's ANT shows that a sufficient condition for a Dedekind ring $R$ to be principle is that it only have finitely many primes. To give an outline of the argument, one starts with a ...
4
votes
1answer
44 views

Discrete valuation on a field - equivalent statements

I have a question and I am stuck, although it should not be too difficult. We consider $K$ a field, $v$ a discrete valuation on $K$ and $O=\{x \in K:v(x)\geq 0\}$ the valuation ring of $v$. Let ...
2
votes
2answers
53 views

Isomorphism of $\mathcal{O}_K$-modules

I came across the following claim in K Conrad's notes: Let $L/K$ be a finite extension of number fields, For fractional ideals $\mathfrak{a}, \mathfrak{b}$, and $\mathfrak{c}$ of $\mathcal{O}_L$, with ...
2
votes
2answers
836 views

Ideal generated by 3 and $1+\sqrt{-5}$ is not a principal ideal in the ring $\Bbb Z[\sqrt{-5}]$

Show that the ideal generated by 3 and $1+\sqrt{-5}$ is not a principal ideal in the ring $\Bbb Z[\sqrt{-5}]$. I fail to understand how can 3 and $1+\sqrt{-5}$ generate an ideal. ...
1
vote
2answers
86 views

Decomposition of an ideal as a product of two ideals

How to show $$5\mathbb{Z}[\sqrt[3]{2}] = (5, \sqrt[3]{2}+2)(5, (\sqrt[3]{2})^2+3\sqrt[3]{2}-1).$$ Firstly, I think that I can say that $$(5, \sqrt[3]{2}+2)(5, (\sqrt[3]{2})^2+3\sqrt[3]{2}-1)= ...
5
votes
3answers
236 views

Overrings of Dedekind domains as localizations

I am taking an independent study where I organize and present weekly material on algebraic number theory to my professor and receive feedback. Next week I am going to cover some miscellaneous topics, ...
1
vote
1answer
111 views

A question on the Chinese Remainder Theorem

This is a question from Lang's ANT, Thm 2 (ch.7, $\S2$). Let $k$ be a number field and $A$ its adele group. In the proof, Lang states Given $x\in A$, let $m$ be a rational integer such that ...
1
vote
1answer
47 views

A set of prime factors of an integer in $\mathcal{O}_k$

I've got a basic question from Thm 2 (ch.7, $\S2$) of Lang's Algebraic Number Theory. Let $k$ be a number field and $A$ its adele group. Let $S_{\infty}$ be the set of Archimedean absolute values of ...
2
votes
2answers
85 views

Ideal norm in a quadratic field

Let $K=\mathbb{Q}[\sqrt{d}]$ be a quadratic field with discriminant $d_K$, let $\mathfrak{a}=(a,\frac{b-\sqrt{d_K}}{2})$ be an ideal. Does the norm $N(\mathfrak{a})=a$? How to prove it?
5
votes
1answer
127 views

The uniqueness of a special maximal ideal factorization

The following problem is from Michael Artin's Algebra, chapter 12, M.6, unstarred: Let $R$ be a domain, and let $I$ be an ideal that is a product of distinct maximal ideals in two ways, say ...
6
votes
1answer
140 views

Are there Infinite Quotients of Algebraic Extensions of $\mathbb{Z}$?

It is well known that $\mathbb{Z}[a_1, \dots, a_n]/(a)$ is a finite ring if each $a_i$ is an algebraic integer and $a \neq 0.$ I suppose this statement becomes wrong if we just require those ...
1
vote
2answers
105 views

Proving that for certain ring of algebraic integers $R$, $R/bR$ is finite

This is a part of proof I try to understand. The situation is the following: Suppose that $a,b,x,y$ are algebraic integers such that $b \neq 0$ and $ax+by=1$. Set $K:=\mathbb{Q}(a,b,x,y)$ and ...
3
votes
1answer
169 views

Trouble with proving $A$ is an integrally closed domain $\Rightarrow$ $A[t]$ is integrally closed domain

This problem has been bugging me for a while. As was stated in the title, I wish to prove: $A$ is an integrally closed domain $\Rightarrow$ $A[t]$ is integrally closed domain Here's what I have ...
2
votes
3answers
111 views

Why is $\mathbb{Z}[\alpha ]$ not finitely generated as an $\mathbb{Z}$-module?

Assume that $\alpha \in \mathbb{C}$ is an algebraic number which is not an algebraic integer. My question is why $\mathbb{Z}[\alpha]$ is not finitely generated as an $\mathbb{Z}$-module. Clearly there ...
1
vote
0answers
68 views

ideals in rings of algebraic integers are finitely generated

I am trying to write about rings of algebraic integers $\mathcal{O}_K$ in a number field $K$ without introducing to much field theory. I want to show that these rings are Dedekind. First of all I want ...
4
votes
1answer
260 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 ...
6
votes
2answers
173 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
131 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
41 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
66 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
178 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
66 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
99 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 ...