# Tagged Questions

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### probability that two randomly selected integers of an imaginary quadratic field of class number 1 are coprime

Given an imaginary quadratic field $\mathbb{Q}(\sqrt{-D})$, where $D$ is a Heegner number (1, 2, 3, 7, 11, 19, 43, 67, 163), what is the probability that two randomly selected elements of that fields' ...
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### Which is the best book on Goldbach conjecture research

Is there a book which summarizes the major research results in the past, and current research trends, for the Goldbach conjecture? I know, much progress has been made in Analytic Number theory in ...
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### Can anyone recommend an easy to read algebraic number theory book?

Can anyone recommend an easy to read algebraic number theory book ? I prefer a book with good examples. (hints or answers to selected questions if possible. Not sure if it is possible for a book of ...
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### Localization of the Integer Ring

Let $\mathbb{Z}$ be the ring of integers and let $p$ be a prime, then the $p$-localization of $\mathbb{Z}$ is defined as $\mathbb{Z}_{(p)}=\{\displaystyle\frac{a}{b}|a,b\in\mathbb{Z},p\nmid b\}$. I ...
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### Does the data of Galois group, ramified places, and inertia groups, determine a Galois number field?

Suppose I tell you that $K/\mathbb{Q}$ is a finite Galois extension, and I specify the Galois group $G$, and suppose further that I give you a finite list $S$ of places of $\mathbb{Q}$ and for each ...
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### Gentle introduction to algebraic number theory

For context, I am a undergraduate majoring in math. I've taken two semesters of algebra (though I am still a bit shaky on Galois theory). I just finshed a course in elementary number theory which used ...
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### Literature to the ring $\mathbb{Z}[\phi]$ where $\phi=\frac{1+\sqrt{5}}{2}$ is the golden ratio

I know few about algebraic number theory but recently I stumbled upon the ring $\mathbb{Z}[\phi]$ where $\phi = \frac{1+\sqrt{5}}{2}$ is the golden ratio. It seems to be a very interesting object to ...
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### Introductive Book on Modular Forms

I'm looking for an introductive book on Modular Forms and their applications to Algebraic Geometry and Algebraic Number Theory. Some Ideas? Explaining you my prerequisites, I've a good knowledge of ...
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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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### Special privilege enjoyed by Elliptic Curves with Complex Multiplication

I think after reading the title one may understand the intention of me, this question is concerned about the Elliptic curves having a Complex Multiplication. I have been reading many theorems, ( ...
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### Generalizations of Monogenic Fields

We recall that a number field $K$ is called monogenic if there exists some $\alpha \in \mathcal{O}_K$ such that $\mathcal{O}_K = \mathbb{Z}[\alpha]$. If instead we take a tower of finite extensions ...
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### Reference for proof of “Dedekind's Criterion”?

It was mentioned to me briefly in passing about a criterion for rings of integers, referred to as Dedekind's Criterion. The Criterion essentially said that a ring $\mathbb{Z}[\omega]$ (for ...
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### Tamagawa numbers and Genus class numbers

I was reading the paper of Prof.Franz Lemmermeyer titled "Pell-conics" which is here, in that the author writes in page 9 that one can define Tamagawa numbers as  c_p = \begin{cases} 2 & \text{ ...
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### A reference and an explanation needed?

In my previous question I was asking for a method to construct a global point if we have local points with us which is here, but I got an answer, it didn't serve the entire purpose, but later on due ...