Questions related to the algebraic structure of algebraic integers

learn more… | top users | synonyms

1
vote
2answers
277 views

Algebraic Number Theory - Integral Basis

Let $K$ be a number field with $[K:Q] =n$. Let $O_k$ be its ring of algebraic integers. I understand how there is an integral basis for $Q$, i.e. $\exists$ a $Q$-basis of $K$ consisting of elements ...
3
votes
1answer
91 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 ...
3
votes
1answer
107 views

Cyclic extension of local fields

Let $K/k$ be an extension of number fields. Is it true that for almost all places $v$ of $k$, the extension $K \otimes_k k_v / k_v$ is a cyclic extension of local fields? (Maybe under some ...
1
vote
1answer
188 views

Dirichlet Characters modulo $260$

I want to count the number of Dirichlet characters with given properties: Number of Dirichlet characters modulo $260$ Number of quadratic Dirichlet characters modulo $260$ Number of primitive ...
0
votes
1answer
44 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 ...
5
votes
1answer
129 views

Set of locations where the Hilbert symbol is not equal to $1$

Let $V$ be the set of prime together with the symbol $\infty$. For a prime $v=p$, denote the $p$-adic numbers by $\mathbb{Q}_p$ and the real numbers by $\mathbb{Q}_\infty$. For $v\in V$ the Hilbert ...
2
votes
2answers
86 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 ...
16
votes
3answers
507 views

Galois Group of the Hilbert Class Field

Let $K/\mathbb{Q}$ be a number field with Galois group G and let $L/K$ be the Hilbert class field of $K.$ It is easy to show that $L$ is Galois over $\mathbb{Q}$ and I am interested in knowing this ...
5
votes
1answer
168 views

When a number is a square in the p-adic rationals - proof question (Quadratic Residues)

I'm a little stuck with the proof of a theorem I'm trying to understand. The theorem is as follows: "For odd prime $p$, suppose for $\alpha \in Q_{p}$ (the p-adic rationals) that $|\alpha|_p=1$. Then ...
4
votes
1answer
105 views

Why is this a prime ideal?

I am reading the proof in Marcus' number fields of the following fact: Let $p$ be a prime in $\Bbb{Z}$ and suppose $p$ is ramified in a number ring $R$ (of some number field $K$). Then $p | ...
9
votes
2answers
474 views

Exact power of $p$ that divides the discriminant of an algebraic number field

I am doing Marcus problem 21 (b) of chapter 3. The setup for this problem is given in problem 20: Setup: Let $L/K$ be a finite extension of algebraic number fields. Write $R = \mathcal{O}_K$ ...
4
votes
1answer
135 views

Factorization of Ideals in Dedekind Domains Proof

I am trying to understand a proof of the following statement: Let $R$ be a Dedekind ring and let $I$ be an ideal of $R$. $I$ is contained in only a finite number of prime ideals ...
2
votes
1answer
105 views

Fractional ideals in quadratic field extension

I have some problems with the topic "fractional ideals". I have two questions: Compute a generator $\alpha$ of the fractional ideal $\Bbb{Z}+\Bbb{Z}(\phi^3(5+\sqrt{31}))$, thus find ...
3
votes
2answers
132 views

Preservation of being a norm under field extension

I'm reading a paper that purports to prove the proposition: Let $K/E$ be a cyclic extension of CM number fields of degree p (an odd prime number). Let $G$ be the Galois group. Let $t$ be the number ...
3
votes
2answers
122 views

In general how to prove or disprove certain types of ideal?

i've come across a lot of questions recently that ask you whether or not there exist certain kinds of ideal, say; does there exist an ideal$ J $of $\mathbb{Z}[i]$ for which $\mathbb{Z}[i] /J$ is a ...
12
votes
2answers
717 views

Eisenstein and Quadratic Reciprocity as a consequence of Artin Reciprocity, and Composition of Reciprocity Laws

Question 1: I've heard that Eisenstein and Quadratic Reciprocity can be derived from the Artin Reciprocity by applying it to certain field extensions. But I haven't seen on any reference an explicit ...
4
votes
1answer
147 views

Lemma in KCd's notes on Totally Ramified Primes and Eisenstein Polynomials

I am reading Keith Conrad's notes here on Totally Ramified Primes and Eisenstein Polynomials. I am trying to understand the proof of Lemma 3.1 on page 4 of his notes which I will reproduce here: ...
6
votes
2answers
144 views

Alternative proof that $[\Bbb{Q}(\zeta_n) : \Bbb{Q} ]= \varphi(n)$ uses circular reasoning?

I am doing exercise 3.24 of Marcus which is the following. Let $L,K$ be number fields with $L/K$ a finite extension (of degree $[L:K] = n$) with $R = \mathcal{O}_K$ and $S = \mathcal{O}_L$. ...
8
votes
3answers
525 views

Attaining the norm of an ideal in a number field by the norm of an element

Let $K$ be a number field of degree $n$ and $\mathfrak{a}$ be an ideal in its ring of integers $\mathcal{O}_K$. We can consider: The norm $N(\mathfrak{a})$ of $\mathfrak{a}$. The norms $N(x)$ of the ...
4
votes
2answers
190 views

Does this equation have integer solutions

Let $g\geq 2$ be an integer. (It will be the genus of some curve.) Are there positive integers $d$ and $e$ such that the equality $$ (e-2)(e-1) = 2d(g-1)+2$$ holds?
6
votes
1answer
201 views

Why is this isomorphism $M \otimes_K L \stackrel{\simeq}{\longrightarrow} M^{[L:K]}$ an isomorphism of $M$ - algebras?

Suppose that $L/K$ is a finite separable extension of fields and let $M$ denote the Galois closure of $L$. Let $\textrm{Hom}_K(L,M)$ denote the set of all $K$ - algebra homomorphisms from $L$ to $M$. ...
2
votes
1answer
95 views

Factorization of $p\mathcal{O}_K$ in $\mathcal{O}_K$

Let $K=\mathbb{Q}(\alpha)$, where $\alpha$ is a root of $f(x)=x^3+x+1$. If $p$ is a rational prime. What can you say about factorization of $p\mathcal{O}_K$ in $\mathcal{O}_K$? I have this: The ...
4
votes
2answers
134 views

Question on Proposition Neukirch 10.3 - Splitting of prime ideals in $\Bbb{Q}(\zeta_n)$

I am reading Proposition 10.3 of Neukirch which I have appended below: Proposition 10.3 (Neukirch): Let $n = \prod_{p} p^{\nu_p}$ be the factorisation of the positive integer $n$ into prime ...
1
vote
1answer
279 views

Class number/ quadratic field/divisibility-$2$

As I am poor in construction of mathematical problem, I am not getting good answers from MSE members. However, this time I constructed the following problem in best possible way. So, I hope I will get ...
2
votes
1answer
89 views

is the idele class group a flat Z-module?

If $K$ is a local or global field and I define $C_K$ to be the idele class group if $K$ is global and I let $C_K=K^{\times}$ is $K$ is local, then as a $\mathbb{Z}$-module will it be flat? i.e. If $$ ...
2
votes
1answer
297 views

Class number/ quadratic field/divisibility

If we take $n$ an even integer and greater than $5$, then $\mathbb Q(\sqrt{1 - 4k^n})$ are divisible by $n$, other than for $k = 13$ and $n = 8$. Why this is happened? If we take $n$ less than 5 (I ...
4
votes
0answers
146 views

Complex multiplication - Ray class fields

I'm pretty new to complex multiplication and am struggling with Corollary 5.20 in Elliptic Curves with Complex Multiplication and the Conjecture of Birch and Swinnerton-Dyer by Rubin. According to ...
2
votes
2answers
200 views

Ideals of norm 2 in $\mathbb{Z}[\zeta_{n}]$

I have a few questions about ideals the ring of integers $\mathbb{Z}[\zeta_{n}]$ in a cyclotomic number field. Specifically, I'm trying to classify the ideals of norm 2. I know that the Gaussian ...
3
votes
2answers
133 views

Square of the sum of n positive numbers

I have a following problem When we want to write $a^2 + b^2$ in terms of $(a \pm b)^2$ we can do it like that $$a^2 +b^2 = \frac{(a+b)^2}{2} + \frac{(a-b)^2}{2}.$$ Can we do anything similar for ...
3
votes
3answers
323 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 ...
4
votes
2answers
613 views

Prove that the equation $y^2=x^3-73$ has no integer solutions

Prove that there are no integers $x,y$ such that $y^2=x^3-73$. Thank you.
1
vote
1answer
156 views

Ideals as a product of prime ideals

Suppose we are working in $\Bbb{Q}(\sqrt{-41})$. Given a ideal, for example $(2-\sqrt{-41})$ (we especially work in $\Bbb{Z}[\omega_{-41}]$). We know that this is a Dedekind ring, thus we have unique ...
6
votes
2answers
735 views

Are the p-adic integers the ring of integers of the field of p-adic numbers?

This question was much simpler, but as I was typing it, it became a chain of questions. My starting question was Is $\mathbb{Z}_p$ (obtained by the inverse limit procedure with the directed ...
1
vote
1answer
185 views

Prime cyclotomic extension and a quadratic subextension

Let $\zeta \in \mathbb{C}$ a primitive $p^{th}$ root of unit ($\zeta^p=1$ holds and no smaller power works) with $p$ an odd primeand assume $p > 2$. Consider $E = \mathbb{Q}(\zeta)$. This ...
5
votes
1answer
140 views

Alternative integral basis for $\Bbb{Z}[w]$

Write $w= e^{2\pi i/m}$ for $m \geq 3$. Consider the number field $K = \Bbb{Q}(\omega)$ and the ring of integers $\mathcal{O}_K = \Bbb{Z}[w]$ that has the usual integral basis $$B = ...
2
votes
2answers
159 views

Roots of unity in quartic fields

It's a well known fact that the roots of unity of the ring of integers of $\mathbb{Q}(\sqrt d)$ where $d\in \mathbb{Z}$ is squarefree, is $\{\pm 1, \pm i\}$ when $d=-1$; $\{\pm 1, \pm \omega, \pm ...
5
votes
1answer
159 views

Equivalent Definition of Non-Archimedean Local Field

Wikipedia states that there is an equivalent definition of non-archimedean local fields: "it is a field that is complete with respect to a discrete valuation and whose residue field is finite." ...
5
votes
1answer
481 views

Maximal Unramified Extension of $\mathbb{F}_p((t))$

The maximal unramified extension of $\mathbb{Q}_p$ can be described quite explicitly: add all roots of unity of order prime to $p$. This is done by the correspondence between finite unramified ...
1
vote
1answer
90 views

Is the Idele class group Hausdorff?

I was wondering if for a global field (function or number field) $K$, is $C_K$ Hausdorff? Thank you
4
votes
1answer
167 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 ...
3
votes
2answers
168 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 ...
7
votes
2answers
974 views

Integral Basis for Cubic Fields

I'm trying to follow a text (Lang's Algebraic Number Theory) in which it fully determines an integral basis for quadratic fields (also seen here). Is there any easy or analogous way to determine one ...
3
votes
2answers
201 views

Quadratic forms over $\mathbb Q$

How to solve a problem like this: Find out which elements $N \in \mathbb N$ are represented by the quadratic form $\left \langle 2,3,2 \right \rangle$ in $\mathbb Q$. The form is $$ f(x,y,z) = 2 ...
0
votes
0answers
88 views

Proving equation (with Hilbert symbol)

Let $K$ be a field with non-zero elements $a,b,c \in K$ and let $(. , .)$ be the Hilbert symbol. Let $(a,-c)=(-1,ac)$ and $(b,-c)=(-1,bc)$. How to show that $(-ab,-c)=(-1,-abc)$ ?
4
votes
1answer
502 views

Discrete subgroups are lattices

If $V$ is an $n$-dimensional real vector space, a lattice in $V$ is a subgroup of the form $\Gamma=\mathbb{Z}v_1+\dots+\mathbb{Z}v_m$ where $v_1,\dots,v_m\in V$ are are $\mathbb{R}$-linearly ...
6
votes
1answer
391 views

Finding the units in $\mathbb{Z}[\sqrt[3]{2}]$ and other questions

I'm reading a paper in which they solve the equation: $$a^3-2b^3=\pm 1$$ in integers using algebraic number theory. The number $a-b\alpha$, with $\alpha=\sqrt[3]{2}$, is a unit in ...
3
votes
1answer
195 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 ...
2
votes
2answers
284 views

Infinitely many primes in the ring of integers

Let $K$ a number field, such that $\mathcal{O}_K= \mathbb{Z}[\alpha]$ for some $\alpha$ algebraic integer. Prove that there are infinitely many primes $\mathcal{P} \in \mathcal{O}_K$, such that ...
7
votes
1answer
149 views

Are there infinitely many pairs of rational numbers such that…

Are there infinitely many pairs of rational numbers $(a,b)$ such that $a^3+1$ is not a square in $\mathbf{Q}$, $b^3+2$ is not a square in $\mathbf{Q}$ and $b^3+2 = x^2(a^3+1)$ for some $x$ in ...
3
votes
2answers
187 views

Set of sum of cosines is finite

Let $x,y \in \mathbb{R}$ such that the set $\{\cos{(n\pi x)} + \cos{(n\pi y)} | n \in \mathbb{N} \}$ is finite. Show that $x$ and $y$ are rational. I have been trying to consider a graph of this set ...