Questions related to the algebraic structure of algebraic integers

learn more… | top users | synonyms

1
vote
2answers
86 views

Is $p^n\mathbb Z_p\cong \mathbb Z_p$ as additive groups?

Is it true that $p^n\mathbb{Z}_p\cong \mathbb Z_p$ as additive groups? Here $\mathbb Z_p$ is the ring of $p$-adic integers for $p$ prime and $n$ is any positive integer. Thanks
0
votes
2answers
89 views

If $\,p\,$ is prime, is $\,p^n\mathbb{Z}_p=\mathbb{Z}_p\,$ for any positive integer $n$?

Is $\,p^n\mathbb{Z}_p=\mathbb{Z}_p\,$ for any positive integer $n\,?$ $\mathbb{Z}_p =$ ring of $p$-adic integers, $\,p$ prime. Thanks.
2
votes
0answers
78 views

Kronecker's approach to unique factorisation in algebraic number theory: books and references

I have done a short course (one semester) on algebraic number theory at beginning graduate level, in which the Dedekind theory of ideals features prominently. However I have since discovered that ...
7
votes
0answers
87 views

Generalization of Kummer isomorphism?

Let $p$ be a prime number and denote by $\mathbb{F}_p(1)$ the one dimensional vector space over $\mathbb{F}_p$ endowed with an action of $G:=Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p)$ via the mod $p$ ...
1
vote
1answer
193 views

A characterization of the module function on a locally compact division ring

References: Weil's Basic Number Theory(written as BNT). Bourbaki's Commutative Algebra(written as BCA). Let $K$ be a topological ring with an identity. Suppose every non-zero element of $K$ is ...
1
vote
0answers
43 views

Values of virtual characters

Let $G$ be a finite group. Let $K$ be a number field and $K^c\subset\mathbb{C}$ its algebraic (separable) closure. Denote by $R_G$ the additive group of functions generated by the characters of n-...
0
votes
1answer
45 views

Why does a Norm Form have rational coefficients?

[From Wolfgang Schmidt's "Diophantine Approximation and Diophantine Equations"] "Let $K$ be a number field with $[K:\mathbb{Q}] = d$ and $L(X)$ be a linear form, say $L(X) = L(x_1,\dots,x_n) = \...
0
votes
0answers
51 views

Solutions to a periodic modular structure

Given $n$ primes $p_{0},p_{1},\dots,p_{n-1}\in(2^{b},2^{b+1})$ for a fixed $b \geq 4$ such that $\prod_{i=0}^{n-1}p_{i} > 2^{2^{b}}$. Is it possible to find $N,d > 0$ such that for any given ...
1
vote
2answers
228 views

Ray class field and ring class field

Let $K$ be quadratic number field and let $O$ be an order of $K$. A modulus $m$ of $K$ is a formal product $m_0\cdot m_\infty$ of finitely many finite primes $m_0$ and finitely many infinite real ...
3
votes
1answer
298 views

What is a relative integral basis?

Reviewing some number theory, particularly relative discriminants for extensions of number fields, I'm running into a common problem where when I look for the definition of a term: all I can find are ...
2
votes
1answer
65 views

Extensions of number fields

Let $L|K$ be an extension of number fields and consider the corresponding (integral) extension of ring of integers: $R_L|R_K$. Note that $R_L$ and $R_K$ are finitely generated over $\mathbb{Z}$, hence ...
5
votes
1answer
134 views

Ring of integers of a degree $5$ extension

Consider the polynomial $P(X) = X^5 - X + 1 \in \mathbb{Q}[X]$, and let $x \in \mathbb{C}$ be a root of $P(X)$. Let $K = \mathbb{Q}(x)$. How can you prove that the ring of integers $\mathcal{O}_K$ is ...
2
votes
1answer
111 views

Norm map in Ideles

If $L/K$ is a finite extension, then there is a natural norm map from $\mathbf{A}^*_L$ to $\mathbf{A}^*_K$. This is a continuous homomorphism $$\text{N}^L_K: \mathbf{A}^*_L\rightarrow \mathbf{A}^*_K$$ ...
1
vote
0answers
71 views

Why is this power residue symbol $=1$?

I am reading this paper on Fermat's last theorem by Stevenhagen and Lenstra, and am confused with one statement. At line $4$ on page $502$, one finds the statement: As $\left(\dfrac{x}{q}\right)$...
1
vote
2answers
122 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 $R:=...
2
votes
1answer
125 views

Isomorphism of DVR quotient with quotient of completion

I'm confused about the proof of Lemma 7.25 from J.S. Milne's course notes on algebraic number theory, available here. Here is the situation: $A$ is the valuation ring corresponding to some discrete ...
2
votes
1answer
307 views

Questions about integral normal bases for subfields of cyclotomic fields

An element $\theta$ in a Galois extension $L$ of $\mathbb{Q}$ is said to give an integral normal basis if the ring of algebraic integers in $L$ is $\sum \mathbb{Z}\sigma(\theta)$ with $\sigma$ running ...
3
votes
1answer
161 views

Question on integral closure in $\mathbb{Q}[\alpha]$

Let $\alpha$ be a root of $f(x) = x^{3} -2x +6$, $ \ \mathbb{K} = \mathbb{Q}[\alpha]$. Prove that $ O _{\mathbb{K}} = \mathbb{Z[\alpha]}$. What I've done: $f$ is irreducible, so $disc\{1,\alpha, ...
11
votes
1answer
289 views

What does the German word “Zerlegungsautomorphismus” translate to?

I would like to know if any of our German friends can translate that word for me? Zerlegung is factorisation isn't it? So what is factorisation automorphism? This is taken from Deuring's paper “Die ...
8
votes
1answer
425 views

Why do no prime ideals ramify in the extension $\mathbb{Q}(\sqrt{p }, \sqrt{q})/\mathbb{Q}(\sqrt{pq })$?

Let $p,q $ be odd integer primes, $p \equiv 1 \pmod 4$ and $q \equiv 3 \pmod 4$. $K = \mathbb{Q }[\sqrt{pq }]$, $L = \mathbb{Q}[\sqrt{p }, \sqrt{q} ]$. Why a prime ideal in $O_{K}$ doesn't ramify in $...
3
votes
1answer
75 views

$L$-function of character in terms of other character

Let $F/K$ be a finite Galois field extension and $\varphi : \mathfrak{I}_K \rightarrow \mathbb{C}^{\times}$ a Hecke character of $K$.Define $\psi = \varphi \circ N_{F/K}$ as a Hecke character of $F$. ...
14
votes
1answer
192 views

Galois Properties of the Values of Modular Forms

Let $f \in S_0(N)$ be a normalized Hecke eigenform. It is well known that its coefficients are algebraic integers, and $f^\sigma$ lies in $S_0(N)$ for $\sigma \in G_{\mathbb{Q}}$. At CM points $z \in \...
2
votes
0answers
224 views

Number of ideals in a ring of algebraic integers with bounded norm

Let $K$ be a number field and $\mathcal{O}_K$ to be the ring of algebraic integers. I was wondering if there is some sort of asymptotic formula or bound for say number of ideals in $\mathcal{O}_K$, ...
10
votes
1answer
295 views

L-function for Dirichlet characters vs Hecke character

For a Dirichlet character $\chi: \left(\mathbb{Z}/p\mathbb{Z}\right)^{\times} \to \mathbb{C}$, the Dirichlet L function is $$\prod_{q \neq p} (1 - \chi(q)q^{-s})^{-1}$$ If we lift this character to $\...
3
votes
1answer
48 views

Harmonizing local and global definitions of discriminant ideal

I began skimming PLC's 8430 handout 5 (on chebotarev density and global class field theory), and there are two versions of "discriminant" presented. The set up is that $R$ is a Dedekind domain, $K$ ...
8
votes
2answers
144 views

Reference for relation between class number of $\Bbb Q[\sqrt{-p}]$ and partial quotients of $\sqrt p$

So in Ireland and Rosen's, "Classical Introduction to Modern Number Theory", they mention the following incredible fact at the end of Chapter 13, section 1. Suppose $p \neq 3$ and $p \equiv 3 \pmod 4$ ...
1
vote
1answer
60 views

Prove automorphism is trivial

I would like to prove the following: Let $L\subset L'$, where $L'$ is a quadratic extension of $L$, and $\rho\in\text{Aut}(L'/L)$, the automorphism group of $L'$ which fixes $L$. Also, let $\...
8
votes
1answer
545 views

Ramanujan-Nagell Theorem Proof Question

I'm currently working through Stewart and Tall's Algebraic Number Theory. In particular, section 4.9 of this book provides a proof of the Ramanujan-Nagell Theorem, which states that the only integer ...
3
votes
3answers
158 views

Raising a rational integer to a $p$-adic power

Let $p$ be a prime number and $d$ a positive integer. If $n\in \mathbb{Z}_p$ (the ring of $p$-adic integers) how would one define $d^n$? Under what conditions would this be an element of $\mathbb{Z}...
1
vote
0answers
409 views

Index of ideals in rings of integers in number fields

Let $R$ be the ring of integers of a number field, or more generally, a finite index subring of it, and let $P$ be a prime ideal of $R$. Is there exist a good bound for the index $[R:P^n]$ in terms of ...
7
votes
2answers
173 views

Primes inert in quadratic field of class number one

Let $K = \mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic field of class number one (i.e. every ideal in $\mathcal{O}_K$ is principal, i.e. $\mathcal{O}_K$ is a principal ideal domain). Let $d_K$ be ...
15
votes
1answer
731 views

what is a “dévissage” argument?

In Serre's "Local Fields" (Chapter 2, section 2, proposition 3) he proves something about field extensions which he breaks into parts: first he dealt with the separable case, and then with the ...
4
votes
0answers
56 views

Separable polynomial mod $\mathfrak{p}$

Let $K/L$ be a galois extension with $L = K(\alpha)$ and let $f$ be the minimal polynomial of $\alpha$ over $K$. Let $\mathfrak{p}$ be a prime in $\mathcal{O}_K$ and $p$ be the prime number above ...
3
votes
2answers
398 views

Finding exercises in local fields, following Serre's book

I am reading Serre's "Local Fields". I would like to find more exercises to complement my study. So I am looking for, mabye, exercises from a course that followed this book. Do you know of such a ...
2
votes
2answers
75 views

Norms in extended fields

let's have some notation to start with: $K$ is a number field and $L$ is an extension of $K$. Let $\mathfrak{p}$ be a prime ideal in $K$ and let its norm with respect to $K$ be denoted as $N_{\mathbb{...
5
votes
1answer
129 views

Show that an ideal is unramified

See Advanced Topics in elliptic curves for the full question(see also errata: http://www.math.brown.edu/~jhs/ATAEC/ATAECErrata.pdf): 2.30 (pg 184) Given $E/L$ an elliptic curve with complex ...
10
votes
1answer
855 views

Special cases of the Stark-Heegner theorem with simple proofs

The Stark-Heegner theorem states that the ring of integers of the quadratic number field $\mathbb Q(\sqrt{m})$, where $m$ is a squarefree negative integer, is a principal ideal domain, iff $$m\in\{-1,-...
2
votes
1answer
80 views

Discriminant of $fg$

Let $f,g \in \mathbb{Z}[x]$ both monic. Suppose that $\operatorname{Res}(f,g ) \neq 0 $, where $\operatorname{Res}(f,g) $ is the resultant of $f$ and $g$. Is it true that $ \operatorname{discr}(g) | \...
1
vote
1answer
117 views

What is known on bounds comparing $ [\mathcal{O}_K :\mathbb{Z}[\alpha ]] $ to $\text{disc}(K)$, $\alpha$ a unit?

Let $K = \mathbb{Q}(\sqrt{41})$, $\omega = \frac{1}{2}(1+\sqrt{41})$, and $\mathcal{O}_K= \mathbb{Z}[\omega ]$ be the ring of integers of $K$. Let $\alpha = 27 +10 \omega$ be the fundamental unit of $\...
10
votes
4answers
213 views

Is the number of irreducibles in any number field infinite?

Are there infinitely many irreducibles in the ring of integers of any algebraic number field ? I tried to follow the same argument as we usually do for integers. Suppose there are finitely many ...
5
votes
2answers
148 views

Why do we only consider quadratic domains as Euclidean domains with squarefree integers?

I have been reading "Introductory algebraic number theory" by Alaca and Williams, and in the opening chapters they use the quadratic domains $\mathbb{Z}+\mathbb{z}(\sqrt{m})$ for non-square $m$ and $\...
0
votes
1answer
49 views

Proving that a ring has only finitely many prime ideals

Let $D$ be a Dedeking domain, $\mathfrak{i}$ a nonzero ideal of $D$ and let $B=D/\mathfrak{i}$ be the quotient ring. Then $B$ is a noetherian ring, and every prime ideal of $B$ is maximal. I have ...
2
votes
1answer
533 views

Peano Axioms natural numbers, total order, uniqueness of addition and multiplication

Could you tell me how to prove the following? $(1)$ There exists the unique operation of addition : $+ \ : \ \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ such that $n+0=n$ and $n+ \sigma(m) =...
4
votes
1answer
299 views

What are the bounds on the class number of a cyclotomic field with regulator power of 2?

Let $\mathbb{Q}(\zeta_n)$ be the $n$th cyclotomic field with $n$ being a power of $2$. What is the best known asymptotic upper bound on the class number of $\mathbb{Q}(\zeta_n)$ as n grows? Can we ...
0
votes
0answers
402 views

Norm map of unramified extension

Let $K \subset L$ be an abelian unramified extension of local fields. Is it true that norm map $N:\mathcal O_L^*\mapsto \mathcal O_K^*$ is surjective?
4
votes
2answers
94 views

Silverman Adv. Topics example

I would like to refer you to Silverman's Advanced Topics in the Arithmetic of Elliptic Curves example 10.6: Let $D$ be a nonzero integer, $E:y^2=x^3+D$ with complex multiplication by $\mathcal{O}_K$ ...
0
votes
1answer
100 views

Equality of ideals involving products of ideals

Let $A = \sqrt{-6}$, I need to show $\langle 2 \rangle = \langle 2, A\rangle*\langle 2, A\rangle$ and $\langle 3 \rangle = \langle 3, A \rangle*\langle 3, A\rangle$. I am using * to emphasize the ...
10
votes
2answers
1k views

Factoring a number of complex integers?

Say you are given a number (ex: $377$) and you express it in a form that allows you to factor it over the complex integers: Notice, $377 = 16^2 + 11^2$ Thus: $(16 + 11i) $ and $(16 - 11i)$ Are ...
2
votes
1answer
53 views

Convergence in $\mathbb{Z}_p$

Here is my question: Let $\alpha_0, \dots, \alpha_{p-1} \in \mathbb{Z}_p$ be such that $\alpha_i \equiv i \pmod{p}$ for all $i = 0,\dots, p-1$. Show that, for any $x\in \mathbb{Z}_{p}$, you can find ...
3
votes
1answer
460 views

An application of the Weak Approximation theorem - Artin-Whaples Approximation Theorem

Let us recall the weak approximation theorem from Valuation theory in Algebraic Number Theory. Let $K$ be a field, and let $|\cdot|_{1},\dots, |\cdot|_n$ be pairwise non-equivalent nontrivial ...