Questions related to the algebraic structure of algebraic integers

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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
398 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
170 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
716 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
388 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
128 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
842 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
212 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
530 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
298 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
393 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
93 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
455 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 ...
5
votes
2answers
642 views

Is $\sin(1)$ algebraic over $\mathbb{Q}$?

Is $\sin(1)$ algebraic over $\mathbb{Q}$? At the moment I have no idea how to proceed. Could you tell me how to solve it?
1
vote
0answers
51 views

classification of rank $2$ $\mathbb{Z}/p^n\mathbb{Z}$-algebra with invertible discriminant

Let $p$ be a prime number and $n$ be an integer. Let $A$ be an $\mathbb{Z}/p^n\mathbb{Z}$-algebra of rank $2$ whose discriminant is non invertible. In Serre's book lecture on the mordell Weil theorem (...
1
vote
3answers
132 views

Is the endomorphism of $\mathbb{Z}_{p}$ induced by multiplication by $p^{n}$ surjective?

Let $p$ be a prime number. Is it true that $p^{n}\mathbb{Z}_{p}\cong\mathbb{Z}_{p}$ as additive groups for any natural number $n$ and if so, why? Here, $\mathbb{Z}_{p}$ denotes the ring of $p$-adic ...
3
votes
1answer
189 views

bests book of representation theory for algebraic number theorists

I am looking for some of the best books on representation theory for an algebraic number theorists> I would prefer a book that is more number theoretical (e.g, galois representations, p adic ...
3
votes
0answers
49 views

Term for the number ω in a quadratic number field?

Associated to the quadratic field $K = \mathbb{Q}[\sqrt{D}]$ (for $D$ square-free) is a number, denoted $\omega$ in all the discussions I’ve seen, defined by $$\omega = \begin{cases} \frac{-1 + \sqrt{...
14
votes
1answer
267 views

Modular interpretation of modular curves

Let $\Gamma$ be a congruence subgroup of level $N$. What is the modular interpretation of $\Gamma\backslash \mathcal{H}^*$, by that I mean what are the elliptic curves + additional structure it ...
0
votes
1answer
215 views

Another Error in Neukirch's Algebraic Number Theory?

I'm reading Neukirch's Algebraic Number Theory and trying to do the exercises. I think I may have found another error, but am not sure... Exercise 7. In a noetherian ring $R$ in which every ...
2
votes
1answer
84 views

Abelian extensions with squarefree discriminant

Is it true that for all $2 \leq n \in \mathbb N$ we can find an abelian extension of the rationals with squarefree discriminant and degree n? Are there even infinitely many? Some even degrees may be ...
6
votes
1answer
91 views

Reduction of kernel of isogenies in the CM case

Let $F$ be a number field and $E/F$ an elliptic curve with CM by an order $\mathcal{O}$ in a quadratic imaginary field $K$. Let us suppose that $K\subseteq F$. Let $p$ be a prime that splits in $\...
3
votes
2answers
122 views

On the ramification index in Dedekind extensions

Citing from my textbook... Definition If $R\subseteq R'$ are Dedekind domains and $\mathfrak{P}$ is a nonzero prime ideal of $R'$ and $\mathfrak{p}=\mathfrak{P}\cap R$ then the ramification index of $...
8
votes
1answer
129 views

What's special about characteristic 2?

I'm trying to get the big picture of how bilinear forms and quadratic forms relate over fields $F$ with $char(F) = 2$ and fields with $char(F) \neq 2$. What I gather so far is that if $char(F) \neq ...
2
votes
1answer
45 views

What is a place?

In Specialization of Quadratic and Symmetric Bilinear Forms (page: 3) the author writes "Let also $\lambda: K \to L \cup \infty$ be a place, $\mathfrak o = \mathfrak o_\lambda$ the valuation ring ...
0
votes
3answers
85 views

Cyclotomic integers: Why do we have $x^n+y^n=(x+y)(x+\zeta y)\dots (x+\zeta ^{n-1}y)$?

Why do we have the factorization $$x^n+y^n=(x+y)(x+\zeta y)\dots (x+\zeta ^{n-1}y)$$ for $\zeta$ a primitive $n^{\text{th}}$ root of unity where $n$ is an odd prime?
0
votes
1answer
70 views

Product of all ideals of prime norm

I am unsure about the truth of the answer in How to find all the ideals of a given norm?, where it is claimed "The ideal (p) is the product of all of the prime ideals of norm a power of p (with ...
6
votes
1answer
198 views

What is the integral homology of $\mathrm{GL}_2(\mathbb{Z}[i])$?

I am currently trying to compute homology groups of general linear groups over the ring of integers of an imaginary quadratic number field. As I would like to check my results I would like to know if ...
4
votes
1answer
265 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 ...
3
votes
1answer
183 views

Number of ideals of a given norm

Is there any analytic expression(those involving exp, sin, cos etc...) which gives the number of ideals of a given norm? of course we are lying in an algebraic number field.
12
votes
1answer
2k views

What's a BETTER way to see the Gauss's composition law for binary quadratic forms?

There is a group structure of binary quadratic forms of given discriminant $d$: Let $[f]=[(a,b,c)], [f']=[(a',b',c')],$ where $d=b^2-4 a c=b'^2-4 a' c'.$ The composition of two binary quadratic ...
2
votes
1answer
269 views

Functional equation for Hecke $L$-series

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves, Theorem II.10.3, we have Let $L(s,\psi)$ be the Hecke $L$-series attached to the Größencharakter $\psi$. Then $L(s,\psi)$ has ...
4
votes
1answer
380 views

Error in Neukirch's “Algebraic Number Theory”?

I found what I believe is an error in Neukirch's book, in Chapter 1 Section 3 (Ideals). Exercise 5 states The quotient ring $\mathcal{O}/\mathfrak{a}$ of a dedekind domain by an ideal $\mathfrak{...
3
votes
1answer
135 views

Ring of integers of $K=\Bbb Q[u]$ where $u=\sqrt[3]{p^2q}$

Let $p,q$ be distinct prime numbers $\ge 5$ such that $pq^2 \not\equiv 1\mod9$. Let $K=\Bbb Q[u]$ where $u=\sqrt[3]{p^2q}$, and $A$ be the ring of integers of $K$. I have shown that $u,v=pqu^{-1}\in A$...
4
votes
1answer
58 views

Gauss sums ray class group

In Neukirch's book, page 503, in remark 1 he says that He gives a reference for that. I'm not able to get it. Could someone at least draw the shape of the indicated Gauss sum, beause I have ...
3
votes
1answer
673 views

How do we find the prime ideals of a ring of integers of a number field?

For example, for $F=\mathbb Q(\sqrt{-5})$ the ring of integers of $F$ is $\mathbb Z[\sqrt{-5}]$ (since $-5\equiv3 \pmod 4$). How can we determine the prime ideals of this ring? Another problem is the ...
2
votes
0answers
174 views

Norm of a $\mathbb{Z}/n\mathbb{Z}$-algebra is surjective

Let $n$ be positive integer and $R = \mathbb{Z}[\alpha]$ be the ring of integer of a quadratic number field where $\alpha$ is the root of the quadratic polynomial $X^2 -uX + v \in \mathbb{Z}[X]$, such ...
8
votes
1answer
393 views

Principal ideal domain not euclidean

Can anyone give an example of a principal ideal domain that is not Euclidean and is not isomorphic to $\mathbb{Z}[\frac{1+\sqrt{-a}}{2}]$, $a = 19,43,67,163$? I believe it is conjectured that no ...
1
vote
1answer
55 views

definition of discriminant and traces of number field.

Let $K=\Bbb Q [x]$ be a number field, $A$ be the ring of integers of $K$. Let $(x_1,\cdots,x_n)\in A^n$. In usual, what does it mean $D(x_1,\cdots,x_n)$? Either $\det(Tr_{\Bbb K/ \Bbb Q} (x_ix_j))$ or ...
5
votes
1answer
288 views

Reduction modulo $p$ in number fields

For every prime number $p$, there exist a map $$f:\mathbb{P}^n(\mathbb{Q})\to\mathbb{P}^n(\mathbb{F}_p)$$ defined by: for $P\in \mathbb{P}^n(\mathbb{Q})$, we can find a unique tuple $(x_1,\dots,...
1
vote
1answer
186 views

Ring of integers of $f(X)=X^3+X^2-2X+8$ is principal

Let $K=\Bbb Q[x]$ be a cubic number field with $x$ be a root of $f(X)=X^3+X^2-2X+8$. I want to show that $A$, the ring of integers of $K$ is principal. What I have shown is that $A=\Bbb Z+\Bbb Zx+\...