Questions tagged [algebraic-number-theory]
Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces
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How small can a sum of a few roots of unity be?
Let $n$ be a large natural number, and let $z_1, \ldots, z_{10}$ be (say) ten $n^{th}$ roots of unity: $z_1^n = \ldots = z_{10}^n = 1$. Suppose that the sum $S = z_1+\ldots+z_{10}$ is non-zero. How ...
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10
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"Understanding" $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$
I have heard people say that a major goal of number theory is to understand the absolute Galois group of the rational numbers $G = \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. What do people mean when ...
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2
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Please check my 6-line proof of Fermat's Last Theorem.
Kidding, kidding. But I do have a question about an $n$-line outline of a proof of the first case of FLT, with $n$ relatively small.
Here's a result of Eichler (remark after Theorem 6.23 in ...
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3
answers
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Is there a "purely algebraic" proof of the finiteness of the class number?
The background is as follows: I have been whittling away at my commutative algebra notes (or, rather at commutative algebra itself, I suppose) recently for the occasion of a course I will be teaching ...
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66
votes
7
answers
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How to picture $\mathbb{C}_p$?
I hope this is appropriate for mathoverflow. Understanding $\mathbb{C}_p$ has always been something of a stumbling block for me. A standard thing to do in number theory is to take the completion $\...
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63
votes
3
answers
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Class field theory - a "dead end"?
I found the claim in the title a bit astonishing when I first read it recently in an interview with Michael Rapoport in the German magazine Spiegel (8 February 2019). And I was wondering how he comes ...
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59
votes
4
answers
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Has Fermat's Last Theorem per se been used?
There is a long tradition of mathematicians remarking that FLT in itself is a rather isolated claim, attractive only because of its simplicity. And people often note a great thing about current ...
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2
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Does every ring of integers sit inside a monogenic ring of integers?
Given a number field $K/\mathbf{Q}$ whose ring of integers $\mathcal{O}_K$ is, in general, not of the form $\mathbf{Z}[\alpha]$ (not monogenic), does there exist an extension $L/K$ which has $\mathcal{...
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13
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Erratum for Cassels-Froehlich
Edit 25 April 2010: I have a physical copy of the new printing of the book. I can only assume the LMS is now selling it (but have no details).
IMPORTANT EDIT: THE RESULTS ARE IN! Ok, the deadline has ...
49
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5
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If a unitsquare is partitioned into 101 triangles, is the area of one at least 1%?
Update: The answer to the title question is no, as pointed out by Tapio and Willie. I would be more interested in lower bounds.
Monsky's famous theorem with amazingly tricky proof says that if we ...
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48
votes
4
answers
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Fermat's last theorem over larger fields
Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite.
Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite?
Here $\...
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47
votes
14
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Applications of the Cayley-Hamilton theorem
The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own ...
47
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6
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How do we study Iwasawa theory?
What papers should we read to start? What basic knowledge do we need to understand the question? What is this area really about? And what are people researching on it?
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2
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Formal group laws and L-series
Let E be an elliptic curve, let $L(s) = \sum a_n n^{-s}$
denote its L-function, and set
$$ f(x) = \sum a_n \frac{x^n}{n}. $$
Then Honda has observed that
$$ F(X,Y) = f^{-1}(f(X) + f(Y)) $$
defines ...
- 32.2k
46
votes
3
answers
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Class Numbers and 163
This is a bit fluffier of a question than I usually aim for, so apologies in advance if this doesn't pass the smell test for suitability.
Likely my favorite fun fact in all of number theory is the ...
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45
votes
17
answers
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Good algebraic number theory books
I have just finished a master's degree in mathematics and want to learn everything possible about algebraic number fields and especially applications to the generalized Pell equation (my thesis topic),...
45
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3
answers
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What was the relative importance of FLT vs. higher reciprocity laws in Kummer's invention of algebraic number theory?
This question is inspired in part by this answer of Bill Dubuque, in which he remarks that the fairly common belief that Kummer was motivated by FLT to develop his theory of cyclotomic number fields ...
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42
votes
2
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Abel and Galois (and Arnold)
Question Is there a connection between Abel and Galois theories of polynomial equations?
Recall that for every polynomial $p(x)\in \mathbb{Q}[x]$ (say, without the free coefficient), Abel considered ...
user6976
41
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2
answers
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Motivating Lubin-Tate theory
The Lubin-Tate theory gives an amazingly clean and streamlined way of constructing the subfield (usually denoted) $F_\pi\subset F^\mathrm{ab}$ for a local field $F$ fixed by the Artin map associated ...
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41
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2
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What is an infinite prime in algebraic topology?
The links between algebraic topology (stable homotopy theory in particular) and number theory are nowadays abundant and fruitful. In one direction, there is chromatic homotopy theory, exploiting the ...
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3
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On what kind of objects do the Galois groups act?
I am neither number theorist nor algebraic geometer. I am wondering
whether Galois groups of number fields (say the absolute Galois
group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$) act on objects which
...
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40
votes
1
answer
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First correct proof of FLT for exponent 3?
It is well known that Euler gave the first proof of FLT ($x^n + y^n = z^n$ has
no nontrivial integral solutions for $n > 2$) for exponent $n=3$, but that his proof
had gaps (which are not as easily ...
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39
votes
4
answers
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Which number fields are monogenic? and related questions
A number field $K$ is said to be monogenic when $\mathcal{O}_K=\mathbb{Z}[\alpha]$ for some $\alpha\in\mathcal{O}_K$. What is currently known about which $K$ are monogenic? Which are not? From Marcus'...
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votes
1
answer
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Degree 17 number fields ramified only at 2
The number $17$ is the smallest odd number that occurs as the degree of a number field $K/\mathbb{Q}$ for which the only finite prime that ramifies is $2$. The non-existence for $n < 17$ follows ...
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37
votes
6
answers
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Using algebraic geometry to understand class field theory
In Algebraic Number Theory, S. Lang says "[a geometrical approach] allows one to have a much clearer insight into the whole class field theory, since the existence theorem and
the reciprocity law ...
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votes
8
answers
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Practical applications of algebraic number theory?
I'm interested in learning about any applications, the more worldly the better*.
Pointing to a nice reference on the number field sieve, for example, would be fine.
However, let me mention one ...
36
votes
3
answers
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The roots of unity in a tensor product of commutative rings
For $i\in\{1,2\}$ let $A_i$ be a commutative ring with unity whose additive group is free and finitely-generated. Assume that $A_i$ is connected in the sense that $0$ and $1$ are unique solutions of ...
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36
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1
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Artin reciprocity $\implies $ Cubic reciprocity
I asked this on math.SE a few days ago with no reply, so I'm reposting it here. Hope this is not considered too elementary for MO (feel free to close if so).
I'm trying to understand the proof of ...
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35
votes
2
answers
3k
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How to visualize Dirichlet’s unit theorem?
As the question title asks for, how do others "visualize" Dirichlet’s unit theorem? I just think of it as a result in algebraic number theory and not one in algebraic geometry. Bonus points for ...
35
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3
answers
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If Spec Z is like a Riemann surface, what's the analogue of integration along a contour?
Rings of functions on a nonsingular algebraic curve (which, over $\mathbb{C}$, are holomorphic functions on a compact Riemann surface) and rings of integers in number fields are both examples of ...
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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 $\widetilde{\mathbf Q}=\overline{\mathbf Q}$?
My ...
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0
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Is there a rigid analytic geometry proof of the functional equation for the Riemann zeta function?
The adèles $\mathbb A$ arise naturally when considering the Berkovich space $\mathcal M(\mathbb Z)$ of the integers. Namely, they are the stalk $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ ...
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votes
4
answers
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$A_5$-extension of number fields unramified everywhere
So I was having tea with a colleague immensely more talented than myself and we were discussing his teaching algebraic number theory. He told me that he had given a few examples of abelian and ...
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3
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Simple argument regarding sums of two units in a number field?
I wonder if it is possible to show, without using the Schmidt subspace/Roth theorem/Baker's bounds on linear forms in logarithms or other very deep results, that, in a number field, not all integral ...
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1
answer
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Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $
EDIT, 9 March 2014: when I asked this in 2010, I did not have the courage of my convictions, and so did not ask for an if and only if proof, as Kevin Buzzard quite properly pointed out. Such problems ...
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32
votes
3
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Quick proof of the fact that the ring of integers of $\mathbb Q(\zeta_n)$ is $\mathbb Z[\zeta_n]$?
I cannot find a good reference for the proof that the ring of integers in a cyclotomic field $\mathbb{Q}(\zeta_n)$ is $\mathbb{Z}[\zeta_n]$. The proof I usually find does an induction on the number of ...
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2
answers
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Is equation $xy(x+y)=7z^2+1$ solvable in integers?
Do there exist integers $x,y,z$ such that
$$
xy(x+y)=7z^2 + 1 ?
$$
The motivation is simple. Together with Aubrey de Grey, we developed a computer program that incorporates all standard methods we ...
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votes
10
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What are some interesting problems in the intersection of Algebraic Number Theory and Algebraic Topology?
I'm a beginning graduate student and while my background is primarily in algebraic number theory, I've found myself a bit smitten with the subject of algebraic topology recently after only having read ...
30
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5
answers
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Is there an algorithm for determining whether an expression involving nested radicals is rational?
Specifically, consider expressions involving integers, addition, multiplication, division, and nth roots for any positive integer n. Is there an algorithm that can determine whether such an expression ...
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Why aren't there more classifying spaces in number theory?
Much of modern algebraic number theory can be phrased in the framework of group cohomology. (Okay, this is a bit of a stretch -- much of the part of algebraic number theory that I'm interested in...)....
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votes
1
answer
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Do the algebraic integers form a free abelian group?
It is a well-known fact, proved in every introductory textbook on algebraic number theory, that if $K$ is an algebraic number field, i.e. a finite extension of $\mathbb{Q}$, then its ring $\mathcal{O}...
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1
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Ramanujan and algebraic number theory
One out of the almost endless supply of identities discovered by Ramanujan
is the following:
$$ \sqrt[3]{\rule{0pt}{2ex}\sqrt[3]{2}-1} = \sqrt[3]{\frac19} - \sqrt[3]{\frac29} + \sqrt[3]{\frac49}, $$
...
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3
answers
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Is $x^{2k+1} - 7x^2 + 1$ irreducible?
Question. Is the polynomial $x^{2k+1} - 7x^2 + 1$ irreducible over $\mathbb{Q}$ for every positive integer $k$?
It is irreducible for all positive integers $k \leq 800$.
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3
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Intuition for Zagier's theorem for $\zeta_K(2)$
In 1986, Don Zagier generalized Euler's theorem ($\zeta_\mathbb{Q}(2)=\pi ^2 /6$) to an arbitrary number field $K$:
$$\zeta_K(2)=\frac{\pi^{2r+2s}}{\sqrt{|D|}}\times \sum_v c_v A(x_{v,1})...A(x_{v,s})...
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Is pi = log_a(b) for some integers a, b > 1?
Are there integers $a, b > 1$ such that $\pi = \log_a(b)$?
Or equivalently: are there integers $a,b > 1$ such that $a^\pi = b$?
Note that the transcendence of $\pi$ makes this a problem - ...
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1
answer
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How would Hilbert and Weber think about the Langlands programme?
Explanations to a general mathematical audience about the Langlands programme often advertise it as "non-abelian class field theory". They usually begin as follows: a modern style formulation of ...
28
votes
9
answers
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Suggestions for good books on class field theory
Recently I tried to learn class field theory, but I find it is difficult. I have read the book "Algebraic Number Theory" by J. W. S. Cassels and A. Frohlich. In the book, the approach to class field ...
28
votes
1
answer
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What's special about the circle problem?
Let $K$ be a number field, and let
$$\zeta_{K}(s):= \sum_{0
\neq I \text{ ideal of }O_K} \frac{1}{N_{K/\mathbb{Q}}(I)^s} = \sum_{n \ge 1} \frac{a_n}{n^s}$$
be the Dedekind zeta function of $K$. The ...
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28
votes
6
answers
2k
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Patterns among integer-distance points
Mark each point of $\mathbb{N}^2$ ($\mathbb{N}$ the natural numbers) if its
Euclidean distance from the origin is an integer. One obtains a plot like this, symmetric about the $45^\circ$ diagonal.
...
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votes
1
answer
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Relation between Schanuel's theorem and class number equation
(Crossposted on math stack exchange: https://math.stackexchange.com/questions/4040249/relation-between-schanuels-theorem-and-class-number-equation)
It was recently brought to my attention that there ...
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