# Tagged Questions

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### Solved Problems in Algebraic Number Theory

I'm not too sure whether this is the right place to ask this (and please correct me if it is not), but I'm currently studying a course in Algebraic Number Theory and would like to be pointed in the ...
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### Theory of irrationalities- Faddeev's book

Does anyone know where (if available) I can get a free access to Delone, B. N., Faddeev, D. K., ''The theory of irrationalities of the third degree'' Transl. Math. Monographs 10, Amer. Math. Soc., ...
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### How to proceed doing number theory?

I'm an undergrad majoring in mathematics. Being in first year I'm still exploring new branches of mathematics and till now, It is analysis and Number theory that I've come to have a great interest ...
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### Forcing the discriminant of an integral basis to be a Carmichael number.

I was thinking about the following lemma recently. Lemma: Let $K=\mathbb{Q}(\theta)$ for some algebraic number $\theta$ and let $n=[K:\mathbb{Q}]$. If $\{\tau_1, \,\dots\,, \tau_n\}$ consists of ...
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### Conceptual reason why a quadratic field has $-1$ as a norm if and only if it is a subfield of a $\mathbb{Z}/4$ extension?

I have convinced myself in a computation-heavy, ad-hoc way that a quadratic extension $K$ of $\mathbb{Q}$ occurs as the unique quadratic subfield of a $\mathbb{Z}/4$-extension of $\mathbb{Q}$ if and ...
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### What's so cool about local compactness?

As I study more algebraic number theory, I hear more and more often about local compactness: locally compact fields, locally compact topological groups, Stone-Čech compactification of locally compact ...
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### What is going on in this degree 8 number field that fails to be a quaternion extension of $\mathbb{Q}$?

This is a soft but very mathematically hands-on question. Hopefully it will be interesting to more than just me. Thanks in advance for your help in thinking clearly about what follows. I have been ...
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### What are some strong PhD programs in (algebraic) number theory?

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, ...
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### Are there applications of noncommutative geometry to number theory?

The marriage of algebraic geometry and number theory was celebrated in the twentieth century by the school of Grothendieck. As a consequence, number theory has been completely transformed. On the ...
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### Tate's Thesis: Meaning of Local Functional Equation

I am studying the development of Tate's Thesis in Lang's Algebraic Number Theory and have a conceptual question. The setting: Let $k=\mathbb{Q}_p$. Let $\mu$ be the unique Haar measure giving ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### What are examples of unexpected algebraic numbers of high degree occured in some math problems?

Recently I asked a question about a possible transcendence of the number ...
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### Are algebraic numbers analogous to group elements with finite order?

Would you say that the "elements with finite order" in group theory are analogous to "algebraic numbers" in field theory? I thought this is the case since requiring an algebraic number $\alpha$ to be ...
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### Applications of prime-number theorem in algebraic number theory?

Dirichlet arithmetic progression theorem, or more generally, Chabotarev density theorem, has applications to algebraic number theory, especially in class-field theory. Since we might think of the ...
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### Classgroup of $\mathbb{Q}(\sqrt{2},\sqrt{-13})$

How would you compute the classgroup of the biquadratic number field $\mathbb{Q}(\sqrt{2},\sqrt{-13})$? I would prefer a method as "from scratch" as possible. Please avoid, if possible, quoting ...
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### Are Different Areas of Number Theory That Different

I was wondering if number theorists are "number theorists," or eventually resolve themselves into one of the various branches - i.e., algebraic, analytic, etc. Also out of curiosity, I was wondering ...
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### 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 ...
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### Consequences of the Langlands program

I have been reading the book Fearless Symmetry by Ash and Gross.It talks about Langlands program, which it says is the conjecture that there is a correspondence between any Galois representation ...
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### 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 ...