1
vote
1answer
34 views

Contracted ideals in number fields

I am trying to translate a section of Wolfgang Krull's report "Idealtheorie". At one point (Section $7$ on Quotient Rings) I believe that he makes something like the following statement: Suppose for ...
1
vote
1answer
31 views

Issue in the first French edition of Serre's local fields

I've been reading Serre's Corps Locaux, and I believe my copy is a first edition, as there's only one copyright date listed, 1968. I believe I found an issue on page 57, which (if you're looking at ...
2
votes
1answer
73 views

what are the “points” of the scheme $\mathbb{Z}_8[x] /(x^2 + 7)$

I noticed modulo 8 the quadratic $x^2 + 7$ is zero for four separate values $x = 1,3,5,7 \in \mathbb{Z}_8$. The number of zeros exceeds the degree. I would like to define the "variety" ...
1
vote
1answer
43 views

Factorization of an ideal in a number field

The notes I read gives following technique to factor an ideal in a number field without explanation. Can anyone explain how this technique works? To factor the ideal $(2)$ in $\mathbb{Z}[\sqrt{-5}]$, ...
0
votes
1answer
54 views

Systems of linear equations over integers modulo n

Let $\mathbb Z_n$ be the ring of integers modulo $n$. Let $A\in M_k(\mathbb Z_n)$ be a square matrix of size $k$. Let $X=[x_1, \ldots, x_k]^T$, where $x_i\in\mathbb Z_n$. There is some method to ...
0
votes
0answers
70 views

System of congruences that do not satisfy CRT assumptions (via algorithm)

Let $x_i,a_i\!\in\!\mathbb{Z}$. The following procedure solves a system of congruences $$x \equiv x_i\pmod{a_i}\;\;\text{ for }i\!=\!1,\ldots,n$$ when $a_i$ are pairwise coprime. Assume that ...
1
vote
2answers
42 views

determine the radical $\sqrt{m\mathbb{Z}}$

Task: determine the radical $\sqrt{m\mathbb{Z}}$. Obviously $m\mathbb{Z} \subseteq \sqrt{m\mathbb{Z}}$. But how to systematically determine the elements of $\sqrt{m\mathbb{Z}}$? I guess it has ...
1
vote
1answer
107 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
197 views

Factoring 1001 in $\Bbb Z[\sqrt 7]$

I am solving the problem of factoring 1001 into prime elements in $\Bbb Z[\sqrt 7]$. I have a couple of questions regarding this. It seems that $\Bbb Z[\sqrt 7]$ is an Euclidean domain. But I do ...
1
vote
2answers
107 views

Simple Combinatorics in finite rings

Let $g = [g_{1} g_{2} \dots g_{r}] \in \Bbb Z_{q}^{*r}$ be a given vector with each $g_{i} \in \Bbb Z_{q}^{*}$ where $\Bbb Z_{q}^{*}$ is $\Bbb Z_{q} \backslash \{0\}$ and $q > 6$ is odd. How many ...
0
votes
2answers
61 views

The action of a Galois group on a prime ideal of a Dedekind domain

This is a slight variant of a question I asked earlier. Let $A$ be a commutative Dedekind domain and $K$ its field of fractions. Let $L/K$ be a finite Galois extension with Galois group $G$ and let ...
2
votes
0answers
59 views

The action of a Galois group on a prime ideal in a Dedekind domain

Let $A$ be a commutative Dedekind domain and $K$ its field of fractions. Let $L/K$ be a finite Galois extension with Galois group $G$ and let $B$ be the integral closure of $A$ in $L$. If ...
6
votes
1answer
70 views

Function field question from Silverman's AEC

Just before Proposition 1.7 on page 5 of AEC (2nd ed), Silverman defines $M_P$ as an ideal in the affine coordinate ring. Then he states Proposition 1.7 (the intrinsic characterization of ...
3
votes
2answers
64 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 ...
2
votes
2answers
269 views

Factorization of ideals in $\mathbb{Z}[\sqrt{5}]$

Consider the ring $R=\mathbb{Z}[\sqrt{5}]$. Let $I$ be the following ideal of $R$: $$I:=(3,1+\sqrt{5})$$ My teacher said that the following equation holds: $$I^2=(3)I,$$ but I actually can't ...
4
votes
1answer
97 views

quadratic extension of $\mathbb{Q}(X)$

Consider the ring $\mathbb{Q}[X]$ of polynomials in $X$ with coefficients in the field of rational numbers. Consider the quotient field $\mathbb{Q}(X)$ and let $K$ be the quadratic extension of ...
-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 + ...
9
votes
1answer
276 views

Primes in a Power series ring

Let $\mathbb Z$ be the ring of rational integers. Consider the power series ring $\mathbb Z[[x]]$. It is known that $\mathbb Z[[x]]$ is unique factorization domain. What are the primes in $\mathbb ...
3
votes
0answers
104 views

Hilbert symbol over a ring

Normally the Hilbert symbol over a field $\mathbb{F}$ is defined for $a,b\in\mathbb{F}^*$ as follows: $$ (a,b)=\begin{cases}1,&\text{ if }z^2=ax^2+by^2\text{ has a non-zero solution }(x,y,z)\in ...
2
votes
1answer
179 views

The Picard group of a product of rings.

Reading a book on Theory of Modules, i have found the assertion ${\bf Pic}(A\times B)\cong {\bf Pic}(A)\times {\bf Pic}(B),$ where $A$ and $B$ are commutative rings with unity. I think that the ...
2
votes
1answer
54 views

How many prime ideals are fixed by a given permutation?

Suppose $L$ is a finite Galois field extension of the rational number field $\mathbb{Q}$, and $B$ is the integral closure of $\mathbb{Z}$ in $L$. Let $\sigma$ be an element of the Galois group ...
3
votes
1answer
84 views

Is the ring $(Z_p[[X]] \otimes Q_p)/(X-p)^r$ principal?

Consider the ring $\mathbb{Z}_p[[X]] \otimes_\mathbb{Z} \mathbb{Q}_p$ and the ideal generated by $(X-p)^r$ (for some integer $r$). Is the following true : for all integer $r$, the ring $$ ...
7
votes
2answers
363 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 ...
2
votes
1answer
97 views

Example of a non-free module of a Dedekind domain R that can be embedded in a finite dimensional vector space over K=frac(R)

Let $R$ be a Dedekind domain, $K$ its field of fractions, and $U$ a finite-dimensional vector space over $K$. Let $M$ be an $R$-submodule of $U$ that contains a basis of $U$ (so $M$ "spans" $U$). ...
11
votes
2answers
300 views

Why is $O_K\otimes \mathbb{Z}_p\cong \oplus_{\mathfrak{p}|p}O_{K,\mathfrak{p}}$?

In my old number theory notebook this is stated as a fact. However, I ran into problems when I tried to prove it. First let me state the (supposed) theorem accurately: Theorem (?) Let $K$ be a ...
10
votes
1answer
371 views

Are all subrings of the rationals Euclidean domains?

This is a purely recreational question -- I came up with it when setting an undergraduate example sheet. Let's go with Wikipedia's definition of a Euclidean domain. So an ID $R$ is a Euclidean domain ...
0
votes
1answer
666 views

Dimension of the vector space of homogeneous polynomials.

Let $R$ be a polynomial ring with $n_k$ variables of degree $k$, for $1\leq k\leq m$. Is there a writeable formula to express the dimension of the vector space $R_l$ of degree $l$ homogeneous ...
2
votes
1answer
124 views

A property of different in Dedekind domains

Let $A \subseteq B$ be a finite extension of Dedekind domains such that the extension $K \subseteq L$ of their quotient fields is separable. Let $\mathfrak{p}$ be a maximal ideal of $A$ and let ...
6
votes
1answer
291 views

Integral closure of p-adic integers in maximal unramified extension

Let $\mathbb Q_p$ be the field of p-adic numbers, and let $\mathbb Q_p^{\text{unr}}$ be maximal unramified extension in some algebraic closure of $\mathbb Q_p$. My understanding is that $\mathbb ...
0
votes
1answer
65 views

Dimensions of modules of the maximal compact subrings of locally compact fields

I have checked the list of similar titles, proposed by the site. I hope this is not a repetition. This question arises from a proof of a proposition in the book Basic Number Theory, as follows. ...
15
votes
3answers
462 views

Can a prime in a Dedekind domain be contained in the union of the other prime ideals?

Suppose $R$ is a Dedekind domain with a infinite number of prime ideals. Let $P$ be one of the nonzero prime ideals, and let $U$ be the union of all the other prime ideals except $P$. Is it possible ...
5
votes
2answers
644 views

Find primitive element such that conductor is relatively prime to an ideal (exercise from Neukirch)

This is an exercise from Neukirch, "Algebraic Number Theory", Ch I, Sec 8, Exercise 2, pg 52. It really has me stumped. Suppose $A$ is a Dedekind domain, $K$ its field of fractions, $L$ a finite, ...
9
votes
2answers
1k views

How to check whether an ideal is a prime (or maximal) ideal?

I have a ring $R$ which is known to be a Dedekind domain, but not necessarily a Euclidian domain, and a nonzero ideal generated by one or two elements in this ring. How can I check if this ideal is a ...
1
vote
2answers
244 views

A characterisation of tame ramification

The following is the statement from Algebraic Number Theory by Neukirch (Chapter 2 Proposition(7.7) p155) Blockquote Suppose $K$ is Henselian field, $p=char(\kappa)$ , the character of the ...
1
vote
1answer
286 views

Invertibility of prime ideals in a number ring lying over prime numbers

I have trouble understanding an argument in the proof of the Kummer-Dedekind theorem. I am referring to a proof given in Peter Stevenhagen's notes. http://websites.math.leidenuniv.nl/algebra/ant.pdf ...