Questions related to the algebraic structure of algebraic integers

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3answers
204 views

likely open number theory problem: finite sum of $\zeta(2)$ equal to a square of rationals

Which $n$ can let $S=1+\frac14+\frac19+\cdots+\frac1{n^2}$ be a square of a rational number? Obviously, $1$ and $3$ work, but how to prove they are the only ones? I think this problem is really hard. ...
10
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1answer
352 views

How to determine all valuations of the field $\mathbb{Q(\sqrt[n]{2})}$?

This is an exercise from the book Algebraic Number Theory by Jurgen Neukirch, on page 166. And, after solving several previous exercises, I found this to be particularly difficult to solve. I am ...
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3answers
138 views

Prove that $2$, $3$, $1+ \sqrt{-5}$, and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$.

So the Norm for an element $\alpha = a + b\sqrt{-5}$ in $\mathbb{Z}[\sqrt{-5}]$ is defined as $N(\alpha) = a^2 + 5b^2$ and so i argue by contradiction assume there exists $\alpha$ such that $N(\alpha) ...
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4answers
177 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 ...
10
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1answer
223 views

Galois representations and normal bases

I am not very familiar with the theory of Galois representations, but I do know a bit about both Galois theory and representation theory. Recently I learned about the notion of a normal basis for a ...
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2answers
493 views

Cubic polynomials that generate the same extension?

For quadratic extensions we can easily determine when $\mathbb{Q}(\sqrt{a})=\mathbb{Q}(\sqrt{b})$ by checking if $a/b$ is a square and this is easy to prove. I was wondering if there are any good ...
10
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1answer
231 views

Connection between ramification in number fields and Clifford theory

Consider algebraic number fields $\mathbb{Q} \subseteq K \subseteq L$ with rings of integers $\mathbb{Z}\subseteq \mathcal{O}_K \subseteq \mathcal{O}_L$. If $0 \neq \mathfrak{p} \trianglelefteq ...
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1answer
306 views

What are the branches of the $p$-adic zeta function?

I'm reading the book $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz. In it, Koblitz wants to iterpolate the Riemann Zeta function for the values $\zeta_p(1-k)$ with $k \in ...
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0answers
94 views

If $p\equiv 1,9 \pmod{20}$ is a prime number, then there exist $a,b\in\mathbb{Z}$ such that $p=a^{2}+5b^{2}$.

I have to prove that if $p\equiv 1,9 \pmod{20}$ is a prime number then there exist $a,b\in\mathbb{Z}$ such that $p=a^{2}+5b^{2}$. I consider the quadratic field $\mathbb{Q}(\sqrt{-5})$, with ring of ...
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2answers
135 views

Sets of Prime Numbers Generated By an Irreducible Monic Polynomial

Given a non-constant integral irreducible monic polynomial $f(x)$, the prime factors of its value at integers $x\in\mathbb{N}$ forms a set $\mathcal{P}(f)$. Is it possible that ...
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0answers
481 views

extension of Euler's totient function to number fields

It is well known that the Euler totient function $\varphi$ satisfies the formula $n = \sum_{d | n}\varphi(d)$. This follows for example from the fact that $\mathbb Z / n \mathbb Z$ can be written (as ...
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1answer
992 views

What is a maximal abelian extension of a number field and what does its Galois group look like?

How does one know that a number field $K$ has a maximal abelian extension (unique up to isomorphism) $K^{\text{ab}}$? I've read proofs involving Zorn's lemma that it has an algebraic closure (And ...
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2answers
254 views

Groups of units of $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$

On page 230 of Dummit and Foote's Abstract Algebra, they say: the units of $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$ are determined by the integers $a,b$ with $a^2+ab+b^2=\pm1$ i.e. with ...
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4answers
925 views

(Simple?) applications of Class Field Theory?

Does anyone know any simple/nice applications of class field theory? I would really like to find one related to diophantine equations, but anything you got would be good. Thanks
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2answers
1k views

Ramification in a tower of extensions

I'm trying to make sense of all these theorems related to ramification. I was hoping someone would summarize these results. Assume we have: An extension $L/K$ and a some subextensions ...
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1answer
439 views

Why $p$-adically interpolate?

I'm studying $p$-adic analysis now and particularly $p$-adic interpolation; for example, constructions like $p$-adic $L$-functions (Kubota-Leopoldt style). I'm having some difficulty though, and I'd ...
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3answers
1k views

What are the integers $n$ such that $\mathbb{Z}[\sqrt{n}]$ is integrally closed?

I was recently reading about integral ring extensions. One of the first examples given is that $\mathbb{Z}$ is integrally closed in its quotient field $\mathbb{Q}$. Another is that ...
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2answers
318 views

Are rational numbers + numbers constructed with a root = algebraic numbers?

I'm a math newbie, so an intuitive explanation is the most helpful for me, but don't pull your punches with the formulas, if you feel like it. We can construct the rational numbers using the division ...
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4answers
457 views

Why does the equation $x^2-82y^2=\pm2$ have solutions in every $\mathbb{Z}_p$ but not in $\mathbb{Z}$?

I have been working on an exercise in H. P. F. Swinnerton-Dyer's book, A Brief Guide to Algebraic Number Theory. The question is like this: Show that $x^2-82y^2=\pm2$ has solutions in every ...
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2answers
472 views

Primes dividing the values of integer polynomials

Problem: Let $n$ be an integer and $p$ a prime dividing $5(n^2-n+\frac{3}{2})^2-\frac{1}{4}$. Prove that $p \equiv 1 \pmod{10}$. The polynomial can be re-written as ...
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2answers
174 views

$\arctan$ of a square root as a rational multiple of $\pi$

I know that if $x$ is a rational multiple of $\pi$, then $\tan(x)$ is algebraic. Is there a fairly simple way to express $x$ as $\pi\frac{m}{n}$, if $\tan(x)$ is given as a square root of a rational? ...
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1answer
1k 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 ...
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2answers
955 views

Finding the norm in the cyclotomic field $\mathbb{Q}(e^{2\pi i / 5})$

I'm doing one of the exercises of Stewart and Tall's book on Algebraic Number Theory. The problem concerns finding an expression for the norm in the cyclotomic field $K = \mathbb{Q}(e^{2\pi i / 5})$. ...
9
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1answer
510 views

What is the relationship between non-Archimedean places of infinite extensions of number fields and primes in the ring of integers?

Let $K$ be a number field and $L$ an infinite algebraic extension of $K$. Fix a non-trivial absolute value $v$ on $K$ (so $v$ is induced either by an embedding into the complex numbers or by a prime ...
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2answers
259 views

Relationship between different L-functions

What's the relationship between between Artin $L$-functions and Dirichlet or Hecke $L$-functions if $L/K$ is an abelian extension? I've been told that one can interpret the Artin $L$-functions as ...
9
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1answer
163 views

Given a group $G$, does there exist a domain $D$ with $G$ as its ideal class group?

I have only recently encountered algebraic number theory and was wondering if this is the case. If the answer to the question is yes, then can we explicitly construct the domain $D$ ? Since the ...
9
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1answer
863 views

Reference book for Artin-Schreier Theory

The aim of the question is very simple, I would like to study Artin-Schreier Theory, but I have had embarassing difficulties in finding a book which could help me in doing that. In specific I'm ...
9
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1answer
279 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 ...
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1answer
227 views

L-function for Dirichlet characters vs Hecke character

For a Dirichlet character $\chi: \left(\mathbb{Z}/p\mathbb{Z}\right)^{\times} \to \mathbb{C}$, the Dirichlet L function is $$\prod_{q \neq p} (1 - \chi(q)q^{-s})^{-1}$$ If we lift this character to ...
9
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1answer
159 views

Intuitive motivation to try to factor an ideal

In $\mathbb{Z} \sqrt{- 5}$, $2$ is irreducible, but the ideal $(2)$ factors into non-units: $(2)$ = $(2, 1 + \sqrt{- 5})(2, 1 - \sqrt{- 5})$. In general, what gives one the intuitive motivation (or ...
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1answer
498 views

$f'/f\in\mathbb{Z}[[x]]$ for polynomials vs. formal power series $f$

I am curious about the following problem from MIT's Problem Solving Seminar (#26 here, though the link may stop working after a few weeks): Let $f(x) = a_0+a_1x+\cdots\in\mathbb{Z}[[x]]$ be a ...
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2answers
2k views

How to check whether an ideal is a prime (or maximal) ideal?

I have a ring $R$ which is known to be a Dedekind domain, but not necessarily a Euclidian domain, and a nonzero ideal generated by one or two elements in this ring. How can I check if this ideal is a ...
9
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1answer
752 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 ...
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2answers
264 views

What are some equivalent statements of (strong) Goldbach Conjecture?

What are some equivalent statements of (strong) Goldbach Conjecture ? We all know that Riemann Hypothesis has some interesting equivalent statements. My favorites are involved with Mertens ...
9
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1answer
210 views

$\mathbb{Q}(i)$ has no unramified extensions

It is a classical result that every extension of $\mathbb{Q}$ is ramified. Put differently: there are no unramified extensions of $\mathbb{Q}$. The classical proof follows from the following two ...
9
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1answer
85 views

Intersection of class number one fields

Let $F$ and $K$ be two number fields with class number one. How can one prove that the class number of $F \cap K$ is also equal to one. I have been trying to prove something like the intersection of ...
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2answers
124 views

Showing that a real number is an algebraic integer

For what values of $x,y,z\in\mathbb{Z}$, such that $0\leq x,y,z\leq 2, $ the real number $$\alpha:=\frac{1}{3}\left(x+\sqrt[3]{175} \cdot y+\sqrt[3]{245}\cdot z\right)$$ is an algebraic integer i.e. ...
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2answers
440 views

Exact power of $p$ that divides the discriminant of an algebraic number field

I am doing Marcus problem 21 (b) of chapter 3. The setup for this problem is given in problem 20: Setup: Let $L/K$ be a finite extension of algebraic number fields. Write $R = \mathcal{O}_K$ ...
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0answers
629 views

A proof of a theorem on the different in algebraic number fields

Theorem Let $K ⊂ L ⊂ E$ be a tower of algebraic number fields. Suppose that $E/K$ is a Galois extension. Let $B$ and $C$ be the rings of algebraic integers in $L$ and $E$ respectively. Let $G$ be the ...
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5answers
477 views

Applications of class number

There is the notion of class number from algebraic number theory. Why is such a notion defined and what good comes out of it? It is nice if it is $1$; we have unique factorization of all ideals; but ...
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3answers
1k views

Prime ideals in the ring of algebraic integers

Let $\mathcal{O}$ be the ring of all algebraic integers: elements of $\mathbb{C}$ which occur as zeros of monic polynomials with coefficients in $\mathbb{Z}$. It is known that $\mathcal{O}$ is a ...
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1answer
2k views

Reading the mind of Prof. John Coates (motive behind his statement)

To start with the issue, I have been thinking from many days that Birch-Swinnerton-dyer conjectures should have some association with the Galois theory, but one day I got the Article of Tate called as ...
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3answers
627 views

Intuition regarding Chevalley-Warning Theorem

Three versions of the theorem are stated on pages 1-2 in these notes by Pete L. Clark: http://math.uga.edu/~pete/4400ChevalleyWarning.pdf Could anyone offer some intuitive way to think about this ...
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2answers
868 views

What does the discriminant of an algebraic number field mean intuitively?

If $E/F$ is a finite extension of fields and $\alpha_1,\ldots, \alpha_n$ is a basis of $E/F$, the discriminant of $\{\alpha_1,\ldots, \alpha_n\}$ is $$\det(\operatorname{Tr}_{E/F}(\alpha_i\alpha_j))$$ ...
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2answers
451 views

Computing the Hilbert class field

Does anyone know any good source with nice examples of Hilbert class field computations? I'm trying to piece together the theory with some canonical examples.
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3answers
296 views

Solve $x^3+x \equiv 1 \pmod p$

Find all the primes $p$ so that $$x^3+x \equiv 1 \pmod p \tag{1}$$ has integer solutions. We consider $x$ and $y$ are the same solutions iff $x\equiv y \pmod p.$ We can prove that $(1)$ cannot ...
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2answers
785 views

Is the algebraic closure of a $p$-adic field complete

Let $K$ be a finite extension of $\mathbf{Q}_p$, i.e., a $p$-adic field. (Is this standard terminology?) Why is (or why isn't) an algebraic closure $\overline{K}$ complete? Maybe this holds more ...
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2answers
876 views

How can one express $\sqrt{2+\sqrt{2}}$ without using the square root of a square root?

I was trying to review some analysis, and came across problem 3 from page 78 of Walter Rudin's Principles of Mathematical Analysis. As part of the problem, I wanted to try to write ...
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3answers
419 views

Attaining the norm of an ideal in a number field by the norm of an element

Let $K$ be a number field of degree $n$ and $\mathfrak{a}$ be an ideal in its ring of integers $\mathcal{O}_K$. We can consider: The norm $N(\mathfrak{a})$ of $\mathfrak{a}$. The norms $N(x)$ of the ...
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3answers
480 views

Unique factorization in $\mathbb Z(\sqrt{-19})$

An elementary confusion about class number: In $\mathbb Z(\sqrt{-19})$ we have $N(1+\sqrt{-19}) = (1+\sqrt{-19})(1-\sqrt{-19}) = 2^2\cdot 5.$ I see that 2 and 5 are irreducible, 4 is not. In a ...