Questions related to the algebraic structure of algebraic integers

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12
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9answers
216 views

$p = x^2 + xy + y^2$ if and only if $p \equiv 1 \text{ mod }3$?

For a prime number $p \neq 3$, do we have that$$p = x^2 + xy + y^2$$for some $x$, $y \in \mathbb{Z}$ if and only if$$p \equiv 1 \text{ mod }3?$$I suspect this is true from looking at the example$$7 = ...
12
votes
1answer
2k 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 ...
12
votes
1answer
1k 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 ...
12
votes
1answer
289 views

Introduction to the trace formula for people outside number theory

I am looking for references on the trace formula, by which I mean the Selberg trace formula and its successor the Arthur-Selberg trace formula. I am aware that there are "standard references" on the ...
12
votes
2answers
122 views

$L$-function, easiest way to see the following sum?

What is the easiest way to see that$$\sum_{(m, n) \in \mathbb{Z}^2 \setminus \{0, 0\}} (m^2 + n^2)^{-s} = 4\zeta(s)L(s, \chi)?$$Here $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to \...
12
votes
2answers
874 views

A weak converse of $AB=BA\implies e^Ae^B=e^Be^A$ from “Topics in Matrix Analysis” for matrices of algebraic numbers.

It is a well known fact that if $A,B\in M_{n\times n}(\mathbb C)$ and $AB=BA$, then $e^Ae^B=e^Be^A.$ The converse does not hold. Horn and Johnson give the following example in their Topics in Matrix ...
12
votes
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 ...
12
votes
1answer
146 views

When is a number in $\mathbb{Q}(\sqrt{a_1},\ldots,\sqrt{a_n})$?

Given an algebraic number $\alpha$ with minimal polynomial $P(x)$ of degree $2^n$, how can I decide if there are integers $a_1,\ldots,a_n$ such that $\alpha\in\mathbb{Q}(\sqrt{a_1},\ldots,\sqrt{a_n})$?...
11
votes
5answers
2k views

How to tell if a Fibonacci number has an even or odd index

Given only $F_n$, that is the $n$th term of the Fibonacci sequence, how can you tell if $n \equiv 1 \mod 2$ or $n \equiv 0 \mod 2$? I know you can use the Pisano period, however if $n \equiv 1$ or $...
11
votes
1answer
2k 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 ...
11
votes
3answers
658 views

What is so special about negative numbers $m$, $\mathbb{Z}[\sqrt{m}]$?

This question is based on a homework exercise: "Let $m$ be a negative, square-free integer with at least two prime factors. Show that $\mathbb{Z}[\sqrt{m}]$ is not a PID." In an aside comment in the ...
11
votes
1answer
726 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 ...
11
votes
2answers
738 views

Every ideal of an algebraic number field can be principal in a suitable finite extension field

Let $K$ be an algebraic number field. Let $I$ be a non-zero ideal of the ring of integers $\mathcal{O}_K$ in $K$. By class field theory, there exists a finite extension(the Hilbert class field) $L$ of ...
11
votes
3answers
679 views

What is the quotient $\mathbb Z[\sqrt{3}]/(1+2\sqrt{3})$?

I am currently doing a past paper and it asks the following: Prove that for $I=(1+2\sqrt{3})$ we have $\mathbb Z[\sqrt{3}]/I$ a field with $11$ elements. If I assume standard algebraic number ...
11
votes
1answer
583 views

The ring of integers of a number field is finitely generated.

For a number field $K$, we define the ring of integers of $K$ to be $$\mathcal{O}_K:=\{x\in K\big|\ (\exists f\in\mathbb{Z}[X])(f\ \text{ is monic and } f(x)=0)\}.$$ Is there any easy way to see from ...
11
votes
1answer
289 views

What does the German word “Zerlegungsautomorphismus” translate to?

I would like to know if any of our German friends can translate that word for me? Zerlegung is factorisation isn't it? So what is factorisation automorphism? This is taken from Deuring's paper “Die ...
11
votes
2answers
336 views

So what *is* the Euclidean function for $\mathcal{O}_{\mathbb{Q}(\sqrt{69})}$?

It's my understanding that $\mathcal{O}_{\mathbb{Q}(\sqrt{69})}$ is UFD, PID, and Euclidean, but not norm-Euclidean. If it were norm-Euclidean, there would be a solution to $28 = q \left(\frac{5}{2} +...
11
votes
3answers
261 views

The intersection of an infinite number of prime ideals in a ring of integers

Let $\mathcal{O}$ be the ring of integers of a number field, $\{\mathfrak{p}_i,\,i \in \mathbb{N}\}$ a sequence of two-by-two pairwise distinct prime ideals. Does it follow that$$\bigcap_i \mathfrak{p}...
11
votes
1answer
283 views

Elementary proof that $\mathfrak{p}$ unramified in $L,L'$ implies unramified in $LL'$?

Let $K$ be a number field, let $\mathfrak{p}$ be a prime of $K$, and let $L,L'$ be extensions of $K$. Suppose $\mathfrak{p}$ doesn't ramify in either $L$ or $L'$. Is there a simple proof that $\...
11
votes
1answer
695 views

Computing the ring of integers of a number field $K/\mathbb{Q}$: Is $\mathcal{O}_K$ always equal to $\mathbb{Z}[\alpha]$ for some $\alpha$?

This question arose from the need to see the splitting behaviour of primes over an extension of number fields. One criterion is the Kummer's theorem, which gives the decomposition of the base prime in ...
11
votes
2answers
352 views

Motivation on how does complex analysis come to play in number theory?

I am not sure if this is a appropriate question. If it isn't, let me know and I'll delete it. $\textbf{Background}$ I am an undergraduate student and I'm very interested in number theory. I've tried ...
11
votes
1answer
190 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 ...
11
votes
1answer
257 views

If $u=\frac{1+\sqrt5}{2}$, then $u^3=2+\sqrt5$, but $u^2=\frac{3+\sqrt5}{2}$. What is the group that measures the power that makes units look nice?

For $A=\mathbb{Z}[x]/(f)$ with quotient field $K$ and ring of integers $B$, does $U(B)/U(A)$ have a name? For instance $u = \tfrac{1+\sqrt{5}}{2}$ is a unit in $\mathbb{Q}[\sqrt{5}]$, but neither $u$...
11
votes
1answer
64 views

Nondegenerate quadratic form $Q$ in $n$ variables with coefficients in $F_p$, cardinality of $\{x \in (\mathbb{F}_p)^n : Q(x) = 0\}$

There is a book on algebraic number theory I am reading that shows the following theorem. Let $Q$ be a nondegenerate quadratic form in $n$ variables with coefficients in $F_p$ ($p \neq 2$). Then$$\...
11
votes
2answers
552 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 ...
11
votes
0answers
69 views

Affine group, $[L : \mathbb{Q}] = n\varphi(n)$ or ${1\over2}n\varphi(n)$

Let $L$ be the Galois closure of $K = \mathbb{Q}(\sqrt[n]{a})$, where $a \in \mathbb{Q}$, $a > 0$ and suppose $[K : \mathbb{Q}] = n$. How do I see that $[L: \mathbb{Q}] = n\varphi(n)$ or ${1\over2}...
11
votes
0answers
179 views

Class field theory for $p$-groups. (IV.6, exercise 3 from Neukirch's ANT.)

I will use notation as in a preious question of mine. This question is from Neukirch's book "Algebraic number theory," page 305, exercise 3. Notation for the problem Let $G$ be a profinite $p$-...
11
votes
0answers
193 views

Proving that the number of integer solutions of $x^2-Ny^2=1$ is infinite

I am trying to prove that the number of integer solutions of $x^2-Ny^2=1$ is infinite whenever N is a squarefree integer. For this I define norm of $a+b\sqrt N=a^2-Nb^2$. Now I prove that $a+b \sqrt ...
11
votes
0answers
254 views

Irreducibility of cyclotomic polynomials via schemes

A few months ago, someone told me there existed a scheme theoretic proof of the irreducibility of cyclotomic polynomials. I've tried coming up with a proof, and when that didn't really yield anything $...
11
votes
2answers
152 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 $\mathcal{P}(f)\cap\...
10
votes
5answers
1k views

Are numbers of the form $n^2+n+17$ always prime

Someone claimed that a number, multiplied by the number after it plus 17 is always prime, and showed several cases. I'm not a complete amateur in Number Theory, and I know that $17*18+17=17*19$, so it ...
10
votes
3answers
187 views

Something screwy going on in $\mathbb Z[\sqrt{51}]$

In $\mathbb Z[\sqrt{6}]$, I can readily find that $(-1)(2 - \sqrt{6})(2 + \sqrt{6}) = 2$ and $(3 - \sqrt{6})(3 + \sqrt{6}) = 3$. It looks strange but it checks out. But when I try the same thing for $...
10
votes
2answers
1k views

Factoring a number of complex integers?

Say you are given a number (ex: $377$) and you express it in a form that allows you to factor it over the complex integers: Notice, $377 = 16^2 + 11^2$ Thus: $(16 + 11i) $ and $(16 - 11i)$ Are ...
10
votes
2answers
320 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 $(2a+b)^...
10
votes
2answers
771 views

In what senses are archimedean places infinite?

According to Bjorn Poonen's notes here (§2.6), we should add the archimedean places of a number field $K$ to $\operatorname{Spec} \mathscr{O}_K$ in order to get a good analogy with smooth projective ...
10
votes
3answers
2k 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 $\mathbb{Z}[\sqrt{5}...
10
votes
2answers
497 views

Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$

1) Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$. Prove: Is it true that $a \in \mathbb{Q}$? 2) Suppose $a \in \mathbb{C}$, ...
10
votes
2answers
249 views

What are the units in $\mathbb{Z}[\root 3 \of 2]$?

I asked Wolfram Alpha to tell me the fundamental unit of $\mathbb{Z}[\root 3 \of 2]$, it replied $1 - \root 3 \of 2$. Then I tried asking it for $(1 - \root 3 \of 2)^n$ for $-5 \leq n \leq 5$. If I ...
10
votes
2answers
2k 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 $E_1,\ldots,...
10
votes
1answer
286 views

Three angles are linearly independent over $\mathbb{Q}$?

If$$\tan \alpha = 1, \text{ }\tan \beta = {3\over 2}, \text{ }\tan \gamma = 2,$$then does it follow that $\alpha$, $\beta$, $\gamma$ are linearly independent over $\mathbb{Q}$? It is possible to test ...
10
votes
3answers
250 views

Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$?

We know that $f(x)=x^4+1$ is a polynomial irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$. My question is: Is it true that if $f(x)$ has a linear factor over $\...
10
votes
2answers
150 views

What is the minimum polynomial of $x = \sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6} = \cot (7.5^\circ)$?

Inspired by a previous question what let $x = \sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6} = \cot (7.5^\circ)$. What is the minimal polynomial of $x$ ? The theory of algebraic extensions says the degree is $...
10
votes
4answers
523 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 $\...
10
votes
2answers
928 views

Ideal class group of a one-dimensional Noetherian domain

Let $A$ be a one-dimensional Noetherian domain. Let $K$ be its field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is finitely generated $A$-module. It is well-known that B ...
10
votes
2answers
1k views

Value of cyclotomic polynomial evaluated at 1

Let $\Phi_n(x)$ be the usual cyclotomic polynomial (minimal polynomial over the rationals for a primitive nth root of unity). There are many well-known properties, such as $x^n-1 = \Pi_{d|n}\Phi_d(x)$...
10
votes
4answers
219 views

Does the equation $x^2+23y^2=2z^2$ have integer solutions?

I would like to show that the image of the norm map $\text N : \mathbb Z \left[\frac{1 + \sqrt{-23}}{2} \right] \to \mathbb Z$ does not include $2.$ I first thought that the norm map from $\mathbb Q(\...
10
votes
3answers
358 views

Does $\mathbb{F}_p((X))$ has only finitely many extension of a given degree?

We know that $\mathbb{Q}_p$ has only finitely many extensions of a given degree in its algebraic closure. Is it the same for $\mathbb{F}_p((X))$?
10
votes
2answers
323 views

Classifying splittings of primes?

I was wondering what general strategies are available to figure out if a prime splits? I know for quadratic extensions there aren't too many possibilities for how a prime can split, so we essentially ...
10
votes
1answer
2k views

Is $\mathbb Z[\sqrt{-3}]$ Euclidean under some other norm?

I know that $\mathbb Z[\sqrt{-3}]$ is not a Euclidean domain under the usual norm $N(x + y\sqrt{-3}) = x^2 + 3y^2$, but that does not necessarily mean that it can't be a Euclidean domain. Is it ...
10
votes
1answer
712 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 ...