Questions related to the algebraic structure of algebraic integers

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89
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1answer
7k views

What's the significance of Tate's thesis?

I've just sat through several lectures that proved most of the results in Tate's thesis: the self-duality of the adeles, the construction of "zeta functions" by integration, and the proof of the ...
56
votes
4answers
4k views

Fibonacci number that ends with 2014 zeros?

This problem is giving me the hardest time: Prove or disprove that there is a Fibonacci number that ends with 2014 zeros. I tried mathematical induction (for stronger statement that claims that ...
50
votes
5answers
5k views

Are all algebraic integers with absolute value 1 roots of unity?

If we have an algebraic number α with (complex) absolute value 1, it does not follow that α is a root of unity (i.e., that αn=1 for some n). For example, (3/5 + 4/5 i) is not a root of ...
46
votes
5answers
2k views

Golden Number Theory

The Gaussian $\mathbb{Z}[i]$ and Eisenstein $\mathbb{Z}[\omega]$ integers have been used to solve some diophantine equations. I have never seen any examples of the golden integers ...
44
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3answers
1k views
37
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4answers
7k views

The Langlands program for beginners

Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things he must know before he can understand the Langlands program and its ...
35
votes
3answers
718 views

The resemblance between Mordell's theorem and Dirichlet's unit theorem

The first one states that if $E/\mathbf Q$ is an elliptic curve, then $E(\mathbf Q)$ is a finitely generated abelian group. If $K/\mathbf Q$ is a number field, Dirichlet's theorem says (among other ...
31
votes
3answers
2k views

Sums of roots of unity

If the integral linear combination of some $n$th roots of unity has magnitude 1, does this necessarily imply that this linear combination is some root of unity as well? More precisely, let $\zeta_1, ...
31
votes
1answer
668 views

Is my field algebraically closed?

For a field $L$, let $\widetilde L$ be the splitting field of all irreducible polynomials over $L$ having prime-power degree. Question: Do we have $\widetilde{\mathbf Q}=\overline{\mathbf Q}$? ...
30
votes
1answer
421 views

Are elementary and generalized hypergeometric functions sufficient to express all algebraic numbers?

Are (integers) plus (elementary functions) plus (generalized hypergeometric functions) sufficient to represent any algebraic number? For example, the real algebraic number $\alpha\in(-1,0)$ ...
29
votes
4answers
1k views

Fermat's Last Theorem and Kummer's Objection

In 1847 Lamé had announced that he had proven Fermat's Last Theorem. This "proof" was based on the unique factorization in $\mathbb{Z}[e^{2\pi i/p}]$. However, Kummer, proved that when $p=23$ we do ...
29
votes
6answers
578 views

Other interesting consequences of $d=163$?

Question: Any other interesting consequences of $d=163$ having class number $h(-d)=1$ aside from the list below? Let $\tau = \tfrac{1+\sqrt{-163}}{2}$. We have (see notes at end of list), ...
29
votes
4answers
901 views

Can a Mersenne number ever be a Carmichael number?

Can a Mersenne number ever be a Carmichael number? More specifically, can a composite number $m$ of the form $2^n-1$ ever pass the test: $a^{m-1} \equiv 1 \mod m$ for all intergers $a >1$ ...
27
votes
6answers
2k views

Easy way to show that $\mathbb{Z}[\sqrt[3]{2}]$ is the ring of integers of $\mathbb{Q}[\sqrt[3]{2}]$

This seems to be one of those tricky examples. I only know one proof which is quite complicated and follows by localizing $\mathbb{Z}[\sqrt[3]{2}]$ at different primes and then showing it's a DVR. ...
27
votes
1answer
1k views

Relation between the Dedekind Zeta Function and Quadratic Reciprocity

I was trying to learn a little about the Dedekind zeta function. The first place I looked at was obviously the Wikipedia article above. So my question comes from a sentence by the end of the article ...
26
votes
1answer
842 views

Are Primes a Self-Fulfilling Prophecy?

Assume the following process: Let's start with the set of primes $\{p_k\}$ Then we use the Euler product being equivalent to Riemann's Zeta function $$ \prod_{p \text{ prime}} \frac{1}{1-p^{-s}} = ...
25
votes
2answers
551 views

Algebraic numbers that cannot be expressed using integers and elementary functions

Can we give an explicit${^*}$ example of a real algebraic number that provably cannot be represented as an expression built from integers and elementary${^{**}}$ functions only? ${^*}$ explicit ...
23
votes
4answers
760 views

Why is it “easier” to work with function fields than with algebraic number fields?

I just bought a copy of Jürgen Neukirch's book Algebraic Number Theory. While browsing through it I found a section titled § 14. Function Fields in chapter I. In it the author describes ...
23
votes
2answers
866 views

Why is it harder to prove which integers are sums of three squares rather than sums of two squares or four squares?

Background: Let $n$ be an integer and let $p$ be a prime. If $p^{e} || n$, we write $v_{p}(n) = e$. A natural number $n$ is a sum of two integer squares if and only if for each prime $p \equiv 3 \pmod ...
23
votes
2answers
435 views

Are irrational numbers order-isomorphic to real transcendental numbers?

I know that rational numbers are order-isomorphic to real algebraic numbers. Does it imply that irrational numbers are order-isomorphic to real transcendental numbers? I know that the order type of ...
23
votes
1answer
296 views

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$?

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$? I think there are no such polynomials, but how to prove?
22
votes
1answer
670 views

Connectedness of the spectrum of a tensor product.

Let $A$, $B$ be finite free $\mathbb{Z}$-algebras such that $\operatorname{Spec}(A)$ and $\operatorname{Spec}(B)$ are both connected. Is $\operatorname{Spec}(A\otimes_{\mathbb{Z}} B)$ connected?
22
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1answer
377 views

Intuition for Class Numbers

So I've been thinking about the analytic class number formula lately, and class numbers in general and I'm trying to develop a good intuition for them. My basic question, which may be too ...
22
votes
1answer
280 views

Homogeneous forms of degree $n$ in $n$ indeterminates over $\mathbb{Z}$: which ones come from the norm of a number field?

Is there a characterization of the homogeneous forms of degree $n$ in $n$ indeterminates over $\mathbb{Z}$ which occur as the norm of some algebraic number ring with a suitable $\mathbb{Z}$-basis? ...
21
votes
1answer
585 views

Why did Gauss think the reciprocity law so important in number theory?

Gauss's Disquitiones Arithmeticae centers around the quadratic reciprocity law. It seems that he developed the genus theory of integral binary quadratic forms to find a natural proof of the quadratic ...
21
votes
1answer
648 views

Ring of integers in p-adic field

How do we compute the ring of integers in a finite extension of $\mathbb{Q}_p$? Say, for example, in $\mathbb{Q}_p(i)$. Over $\mathbb{Q}$ we would guess $\mathbb{Z}[i]$, compute the discriminant of ...
20
votes
5answers
1k views

Elementary results from Algebraic Number Theory

The purpose of this question is to motivate me to study algebraic number theory. Let me explain. My motivation for studying number theory is to learn about beautiful results with simple, accessible ...
20
votes
2answers
1k views

Unramification of a prime ideal in an order of a finite Galois extension of an algebraic number field

Is the following proposition true? If yes, how would you prove this? Proposition Let $K$ be an algebraic number field. Let $L/K$ be a finite Galois extension. Let $A$ and $B$ be the rings of ...
20
votes
1answer
835 views

Brave New Number Theory

I suppose this is an extremely general question, so I apologize, and perhaps it should be deleted. On the other hand it's an awesome question. Is it clear exactly how much (assumedly algebraic) ...
20
votes
0answers
3k views

What are some strong algebraic number theory PhD programs? [closed]

I am currently applying for PhD programs in the US. My main interests are number theory and algebra. More specifically, I am interested in algebraic number theory (number fields, Galois groups, ...
18
votes
2answers
1k views

Why do we use this definition of “algebraic integer”?

A number is an "algebraic integer" if it is the root to a monic polynomial with integer coefficients. Artin says (Algebra, p. 411): The concept of algebraic integer was one of the most important ...
18
votes
1answer
680 views

Are $\pi$ and $e$ algebraically independent?

Update Edit : Title of this question formerly was "Is there a polynomial relation between $e$ and $\pi$?" Is there a polynomial relation (with algebraic numbers as coefficients) between $e$ or $\pi$ ...
18
votes
1answer
378 views

Class group and factorizations

There is a common characterization of the class group ${\rm Cl}(R)$ as a kind of measure of how badly factorization fails to be unique. The most obvious justification for this sentiment is that the ...
18
votes
0answers
388 views

Which number fields can appear as subfields of a finite-dimensional division algebra over Q with center Q?

I have some idle questions about what's known about finite-dimensional division algebras over $\mathbb{Q}$ (thought of as "noncommutative number fields"). To keep the discussion focused, let's ...
17
votes
3answers
603 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 ...
17
votes
3answers
314 views

What sort of ring is the integral closure of $\mathbb{Z}$ in $\overline{\mathbb{Q}}$?

I begin losing intuition when I start dealing with infinitely generated fields over $\mathbb{Q}$... The naive guess is that it is noetherian of krull dimension 1. Is this correct? A related question ...
17
votes
3answers
308 views

How to prove summation, multiplication, subtraction of two roots of $1+x+\frac{x^2}{2!}+\cdots+\frac{x^p}{p!}=0$ aren't rationals?

Assume $a$, $b$ are distinct roots of the following equation: $$1+x+\frac{x^2}{2!}+\cdots+\frac{x^p}{p!}=0,$$ where $p$ is a prime number and $p \gt 2$. How to prove that $ab$, $a+b$, $a-b$ are not ...
17
votes
2answers
249 views
+50

Without using Heegner-Stark-Baker, $\mathbb{Q}(\sqrt{-11})$ has class number $1$.

Prove that $\mathbb{Q}(\sqrt{-11})$ is of class number $1$. I have found that the ideal $(2)$ of the integer ring $\mathbb{Z}[(1 + \sqrt{-11})/2]$ of $\mathbb{Q}(\sqrt{-11})$ is a prime ideal. ...
17
votes
1answer
409 views

Equilateral polygon in a plane

Let $n$ be a positive integer. Suppose we have an equilateral polygon in the Euclidean plane with the property that all angles except possibly two consecutive ones are an integral multiple of $\pi/n$, ...
16
votes
2answers
208 views
+150

Enumerating Bianchi circles

Background: Katherine Stange describes Schmidt arrangements in "Visualising the arithmetic of imaginary quadratic fields", arXiv:1410.0417. Given an imaginary quadratic field $K$, we study the Bianchi ...
15
votes
3answers
729 views

Finite abelian groups as class groups

Is it known whether every finite abelian group is isomorphic to the ideal class group of the ring of integers in some number field? If so, is it still true if we consider only imaginary quadratic ...
15
votes
3answers
882 views

What is the intuition behind Gauss sums?

Let $ \chi $ be a character on the field $ F_p $, and fix some $a \in F_p $. We define a Gauss sum to be: $g_a (\chi) = \sum_{t\in F_p}\chi(t)\zeta^{at}$ where $\zeta$ is a primitive $p^{th}$ root of ...
15
votes
1answer
427 views

Explicit automorphisms of the field of algebraic numbers

The field $\overline {\bf {Q}} $ of algebraic numbers admits many automorphisms other than conjugation. This follows from Galois theory: the field $\overline {\bf {Q}}$ can be realized as the union ...
15
votes
2answers
586 views

Formula for number of solutions to $x^4+y^4=1$, from Ireland and Rosen #8.18.

There is a sequence of three exercise in Ireland and Rosen's Introduction to Modern Number Theory, Chapter 8, page 106. I can do the first two, but can't finish the third. I can include the proofs to ...
15
votes
2answers
430 views

What would change in mathematics if we knew $\pi+e$ is rational?

It is well known that there's no conclusion now whether $\pi+e$ is rational or not. What would happen if we knew that $\pi+e$ is rational? Specifically, are there related open problems that would be ...
15
votes
2answers
565 views

What is Riemann-Roch in arithmetic all about?

I learn number theory recently and I could not understand what Riemann-Roch was all about in arithmetic; could someone give me a bit hint? What is the advantage of viewing all this stuff geometrically ...
14
votes
2answers
447 views

Why is $(\sqrt{2}+\sqrt{3})^{2008}$ so close to an integer?

Using 5000-digit precision in PARI/GP, I discovered that the fractional part of $(\sqrt{2}+\sqrt{3})^{2008}$ is extremely small, less than $10^{-999}$. Is there a simple explanation for this fact ? ...
14
votes
5answers
728 views

An 'obvious' property of algebraic integers?

I am looking at the book A Brief Guide to Algebraic Number Theory by H. P. F. Swinnerton-Dyer. I like the section on page 1 'the ring of integers' as it gives a motivation for choosing which elements ...
14
votes
2answers
705 views

Ring of integers is a PID but not a Euclidean domain

I have noticed that to prove fields like $\mathbb{Q}(i)$ and $\mathbb{Q}(e^{\frac{2\pi i}{3}})$ have class number one, we show they are Euclidean domains by tessalating the complex plane with the ...
14
votes
2answers
592 views

Why is $\tau(n) \equiv \sigma_{11}(n) \pmod{691}$?

If $n$ is a natural number, let $\displaystyle \sigma_{11}(n) = \sum_{d \mid n} d^{11}$. The modular form $\Delta$ is defined by $\displaystyle \Delta(q) = q \prod_{n=1}^{\infty}(1 - q^n)^{24}$. ...