0
votes
0answers
13 views

Linear combination of prime multiplicities with 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
27 views

$(p)$ is a prime ideal in $\mathbb{Z}[\sqrt{n}]$

Let $n$ be a square-free integer such that $n\equiv 0,2,$ 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
71 views

When does coprimality carry over to the base ring in an 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
47 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
119 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
56 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
86 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
55 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
78 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
77 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 ...
1
vote
0answers
27 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
74 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
79 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
55 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
100 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
83 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
113 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 $a$ ...
0
votes
0answers
68 views

Let $B$ be integral, finite over Dedekind $A$. $\mathcal P$ is the only prime over $p$ iff $B_\mathcal P$ still integral over $A_p$

This appears as a Warning on page 24 of Lang's ANT. Let $B$ be a finite module over a Dedekind ring $A$ and integral over $A$ as a ring. Then $\mathcal P$ is the only prime over $p$ iff $B_\mathcal P$ ...
4
votes
1answer
104 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
38 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
652 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
68 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
210 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
101 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
43 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
63 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
124 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
135 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
143 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
98 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
63 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
212 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
150 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
116 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
38 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
61 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
163 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
62 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
93 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
252 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
80 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
218 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 ...
-2
votes
1answer
146 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 + ...
3
votes
3answers
226 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 a projective $R$-module by producing a short exact sequence that splits, ...
0
votes
0answers
45 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
403 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...