Questions related to the algebraic structure of algebraic integers

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Test on representing prime using eight variate quadratic form

Assume $f_i(x_1,\dots,x_8)$ are linear forms in $\Bbb Z[x_1,\dots,x_8]$ where $i\in\{1,\dots,m\}$. Assume $c_i\in\Bbb Q$ are constants where $i\in\{1,\dots,m\}$. Consider quadratic form ...
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0answers
13 views

Finitely generated modules over principal ideal domain

Let A be principal ideal domain with field of fractions K. L is finite separable extension of K and B is integral closure of A in L. It is obvious that there exists constant d in A, such that d.B is ...
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0answers
20 views

Quartic extension

Consider primes represented by $x^2 + ny^2$. If $m$ and $n$ are two positive integers such that the Legendre symbol $\left(\frac{-m}{n}\right) = 1$, then does there exist a cyclic quartic unramified ...
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1answer
21 views

Integral closure of rings [on hold]

Let A be entire ring with field of fractions K, L is finite field extension of K, B is integral closure of A in L. How to prove that field of fractions of B is L?
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1answer
37 views

How to prove the sum of squares larger than 1/n without induction? [duplicate]

known that: $1\geq R_1 \geq R_2 \geq \dots \geq R_n \geq 0$ and $\sum_{i=1}^n R_i=1$ To prove: $\sum_{i=1}^n R_i^2 \geq \frac{1}{n}$ Using induction, the problem can be easily proved. I'd like to ...
8
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1answer
65 views

Primes of the form $x^2+n\cdot y^2$, given $n$?

In an attempt to get to grips with algebra for a course I intend to follow, I was working through a bunch of exercise sheets. A series of questions got me wondering: Given an integer $n$, is there ...
1
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1answer
31 views

How to find a quadratic form that represents a prime?

Given $a,b,c,p\in\Bbb N$ with $b^2-4ac<0$ and $p$ is a prime with $\bigg(\frac{b^2-4ac}p\bigg)=1$ and $p\nmid b^2-4ac$, then there is at least one primitive form of discrimant $b^2-4ac$ that ...
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2answers
62 views

Why if $\alpha$ is a root of $x^3-5x^2+2$, then $\mathcal O_{\Bbb Q[\alpha]}=\mathbb Z[\alpha]$?

I'm trying to show that if $\alpha$ is a root of the polynomial $x^3-5x^2+2$ and $K=\Bbb Q[\alpha]$, then $\mathcal O_K=\Bbb Z[\alpha]$. This is homework, and one of the previous exercises asks to ...
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0answers
36 views

Prime ideals decomposition

How to prove the decomposition law for prime ideals in finite separable extensions of number fields? (J Neukirch - Algebraic number theory- p47, Proposition 8.3)
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21 views

Arakelov class group

In book Neukirch Algebraic Number theory. Chapter III theorem 1.12. If $K$ is a number field then $O_K^*$ is finitely generated and $cl_K(\mathcal O_K)$ is a finite group. Theorem 1.12 say that ...
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27 views

What are the restrictions in the ramification behavior of a Galois extension of number fields imposed by the Galois group of the extension?

Studying class field theory, I have come across the following Proposition: Proposition. Let $K/E$ be an extension of number fields so that there is no nontrivial unramified subextension $F/E$ with ...
4
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1answer
32 views

Algebraic integers of $\mathbb{Q}(\sqrt{m})$ for $m$ a squarefree integer

I'm currently reading Marcus' "Number Fields," and I'm having difficulty proving the following result: Corollary 2.2: Let $m$ be a squarefree integer. The set of algebraic integers in the quadratic ...
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0answers
31 views

Proof of the structure theorem

I'm reading through Ian Stewart's book "Algebraic Number Theory and Fermat's Last Theorem" (3rd edition) and I'm having trouble with a bit of the proof of Theorem 1.16 (page 29). The part I don't ...
4
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1answer
100 views

Galois invariants of the Tate module of an elliptic curve over a number field

Let $K$ be a number field, $E$ be an elliptic curve over $K$, $l \neq p$ be two different prime numbers and $v$ be a place of $K$ above $l$. I am trying to understand the proof of proposition I.6.7 ...
3
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0answers
33 views

Unit group of an imaginary quadratic ring

Let $R$ be an imaginary quadratic ring. Then, the unit group $R^{\times}$ is finite. To prove this, I worked with normal forms, algebraic integers and the fact that $R \not \subset \mathbb{R}$. But I ...
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1answer
86 views

A Guide to the Proof of Fermat's Last Theorem from the Modularity Theorem

A few years ago, a friend of mine told me that he had taken an advanced undergraduate number theory class (so something that assumed only a knowledge of algebra and mathematical maturity) which ended ...
6
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1answer
116 views

Ramification in $\mathbb Q(i,\sqrt[4]\pi)/\mathbb Q(i)$

Let $\pi\ne1+i$ be a prime element of $\mathbb Z[i]$. I am interested in the ramification in the extension $\mathbb Q(i,\sqrt[4]\pi)/\mathbb Q(i)$, especially over $(1+i)$. I've tried for instance to ...
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0answers
31 views

Decomposition and valuation rings

I am reading Algebraic Number Theory by A. Fröhlich, M. J. Taylor, it first introduced the theory: $(K,u)$ be a field and its absolute value, $(K_u,\bar u)$ be its completion and absolute value ...
5
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1answer
64 views

Dimension of $\mathbb{Q}$-vector spaces $H^m(X, \mathbb{Q})$.

Assume that you can't compute the cohomology group $H^m(X, \mathbb{Q})$ for$$X = \{(x : y : z : w) \in P^3(\mathbb{C}): xy = zw\}$$but you know Weil conjecture. By using Weil conjecture, give the ...
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2answers
89 views

Are these quotient modules isomorphic?

Let $K$ be an algebraic number field and $\mathcal{O}_K$ its ring of integers. For a non-zero ideal $\mathfrak{a}$ of $\mathcal{O}_K$ and an element $c \in \mathcal{O}_K \setminus \{0\}$ I wonder ...
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0answers
31 views

Complete field and field extension.

$(K,u)$ be a pair of the field $K$ and its absolute value $u$, $(K_u, \bar u)$ denotes its completion and the corresponding absolute value. Let $L$ be a field containing $K$, $\pi:K_u\rightarrow ...
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0answers
32 views

Characterization of cosine of rational multiples of $\pi$

Given an algebraic number $x$ such that $-1 \leq x \leq 1$ is there a characterization to figure out whether $\cos^{-1}(x)$ is a rational multiple of $\pi$ or not? One characterization would be that ...
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2answers
125 views

Arithmetic Derivatives: Arithmetic Logarithmic Derivative Problem

In Calculus, whenever we see a constant and want to take the derivative of it, it always is 0. However in Number Theory, we have something called the arithmetic derivative in which we can ...
7
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1answer
47 views

Orbits of the $\text{SL}(n,\mathcal{O}_K)$-action on $\mathbb{P}^{n-1}(K)$ for a number field $K$.

I was reading some notes of Keith Conrad where he proves that the number of orbits of the $\text{SL}(2,\mathcal{O}_K)$-action on $\mathbb{P}^{1}(K)$ for a number field $K$ is precisely the class ...
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0answers
31 views

Unique extension of the absolute value

Let $(K,u)$ be a complete valued field, $u$ be its discrete absolute value (corresponds to a discrete valuation on $K$), then: ($\ast)$ Let $E/K$ is a finite separable field extension, then the ...
3
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1answer
85 views

How to show that the norm of a fractional ideal is well-defined?

Sorry. This might probably be a really easy question, but I am only a beginner in algebraic number theory. So, please bear with me. Let $K$ be an algebraic number field and $\mathcal{O}_K$ its ring ...
5
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0answers
31 views

Are the ring of integers of the constructible numbers a Euclidean domain?

I suspect that since Euclid uses the Euclidean Algorithm to perform division on constructible numbers in Elements, the ring of integers of the constructible numbers are a Euclidean Domain, but I have ...
3
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1answer
40 views

How to “floor” in an imaginary quadratic integer ring?

In his answer to this question What is a concrete example to demonstrate that $\mathcal{O}_{\mathbb{Q}(\sqrt{-19})}$ is NOT a norm-Euclidean domain? Robert Soupe essentially looks up in a map to try ...
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3answers
82 views

Understanding $Gal(\bar k /k)$

According to this book, An Introduction to the Langlands Program, One of the fundamental goals of modern number theory is to understand the Galois group $Gal(\bar k /k)$ where $k$ is a local or a ...
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0answers
24 views

Arakelov Theory and Arakelov curves

There exists a definition of Arakelov Curve in Arakelov theory? My question is because Neukirch (Algebraic Number Theory, Chapter III) defined Arakelov divisors in the set $X=Spec(\mathcal ...
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0answers
32 views

A consequence of “every norm is equivalent to the sup-norm” in a finite dimensional normed vector space

So, I am reading the following proposition in Neukirch's Algebraic Number Theory: Proposition. Let $K$ be a complete field with respect to the valuation $|\:\: |$ and let V be an $n$-dimensional ...
6
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2answers
58 views

unramified quadratic extension of number field

I try to understand the following statement There are only finitely many quadratic unramified extension of a number field $K$ I know by Kummer theory that such extensions are of the form ...
3
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1answer
28 views

Splitting an ideal lying in a prime ideal

An obviously naive question question I can't answer. Let $A$ be a Dedekind domain, $\mathfrak{p}$ a prime ideal of $A$ and $I$ a non-zero ideal lying in $\mathfrak{p}$. The ideal $I$ can be splitted ...
2
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0answers
50 views

Neukirch: Riemann-Roch Theory

In the book Algebraic Number Theory, Chapter III Riemann Roch Theory, theorem 1.12. The two fundamental facts of algebraic number theory, the finiteness of the class number and Dirichlets unit ...
6
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2answers
45 views

In what quadrant should the GCD of two Gaussian integers with both real and imaginary parts be?

Examples: $$\gcd(-9 - 3i, -2 + i)$$ $$\gcd(9 + 3i, -2 - i)$$ $$\gcd(-9 - 3i, -2 - i)$$ $$\gcd(-3 - 9i, -2 + i)$$ $$\gcd(-3 + 9i, -1 + 2i)$$ I've put these through Wolfram Alpha and, for some, ...
2
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1answer
33 views

height of formal group of an elliptic curves

I have an elliptic curve $E$ defined over a complete discrete-valued field $K$ of characteristc $0$. the residue field $k$ is of positive characteristic $p$. Then $E[p]=\mathbb{Z}/p\mathbb{Z} \times ...
7
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1answer
54 views

Two definitions of ramification groups, why are they equivalent?

Let $L|K$ be a finite galois extension and suppose that $v_k$ is a discrete normalized (non-archimedean) valuation of $K$ with positive residue field characteristic $p$, and that $v_K$ admits a unique ...
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3answers
178 views

Given $d \equiv 5 \pmod {10}$, prove $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ never has unique factorization

With the exception of $d = 5$, which gives $\mathbb{Z}[\phi]$, of course (as was explained to me in another question). I'm not concerned about $d$ negative here, though that might provide a clue I ...
2
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0answers
34 views

Which mathematical objects generate the zeroes of $L$-functions?

I've studied analytic and algebraic number theory for years and years, and I encountered a hard question about Riemann zeta function and other kinds of $L$-functions - which might be one of the most ...
2
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1answer
29 views

An abelian number field is either totally real or CM-field

The wikipedia article of totally real number fields says: The totally real number fields play a significant special role in algebraic number theory. An abelian extension of Q is either totally ...
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0answers
13 views

Non-CM totally imaginary number fields

Is there a name for the totally imaginary number fields that are not CM-fields? Any important subclass of number fields with that property, or perhaps a reference where those field are studied in ...
2
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1answer
33 views

Completion and algebraic closure commutable

The following corollary of Krasner´s Lemma says: Let k be a global field and p a prime of k. Then $(\overline{k})_p=\overline{k_p}$. Im wondering if $(\overline{k})_p$ means the completion of ...
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1answer
42 views

Prove that $\mathbb{Z}[\omega + \omega^{-1}]$ is a PID for $\omega = e^{2\pi i /13}$

As the title suggests, I'm trying to prove that $\mathbb{Z}[\omega + \omega^{-1}]$ is a PID for $\omega = e^{2\pi i /13}$. If we define $R := \mathbb{Z}[\omega + \omega^{-1}]$, then $\text{disc}(R) = ...
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1answer
31 views

Is $\widehat{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Q} \cong \mathbb{A}_{\mathbb{Q}}^f$ as topological rings?

Maybe this is rather trivial, but I could not solve this (actually, I think this is not true, however I'm not sure). Is $\widehat{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Q} \cong ...
3
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1answer
71 views

Definition of Selmer-Group for Elliptic Curves

Im facing a problem in Silvermans Book "Arithmetic of elliptic Curves" at the beginning of chapter X.4 concerning the exact sequences. Let $K$ be a number field with a valutaion $v$. I'm ...
0
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1answer
23 views

How to show that $\prod^{p-1}\limits_{j=0} (x+\eta^jy)=x^p+y^p$

How to show that $\prod^{p-1}\limits_{j=0} (x+\eta^jy)=x^p+y^p$, where $p$ is an odd prime, $\eta$ is $p$-th roots of unity and $x,y$ are integers. It could be reduced to the form ...
3
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0answers
34 views

Prove that $\mathbb{A} \cap \mathbb{Q}(\sqrt{2},\sqrt{-3})$ is a PID.

While self-studying algebraic number theory, I came across the following problem: Prove that $\mathbb{A} \cap \mathbb{Q}(\sqrt{2},\sqrt{-3})$ is a PID. where $\mathbb{A} \cap ...
8
votes
3answers
103 views

How to prove there are no solutions to $a^2 - 223 b^2 = -3$.

As the title suggests, I'm trying to prove that there are no solutions to $a^2 - 223b^2 = -3$ (with $a,b\in \mathbb{Z}$). Ordinarily, taking both sides $\mod n$ for some clever choice of $n$ proves ...
2
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1answer
48 views

Two disjoint number fields $K$, $L$ such that $(\mathrm{disc}({\cal O}_K), \mathrm{disc}({\cal O}_L))\neq 1$ but ${\cal O}_L{\cal O}_K={\cal O}_{KL}$

I know that if two disjoint number field $K$, $L$ are such that $(\operatorname{disc}(\mathcal{O}_K), \operatorname{disc}(\mathcal{O}_L))= 1$ then $\mathcal{O}_L\mathcal{O}_K=\mathcal{O}_{KL}$. It is ...
2
votes
1answer
22 views

profinite completion of a ring of integers in a global field

It is well known that the profinite completion of the integers, $\hat{\mathbb{Z}} = \varprojlim \mathbb{Z}/n\mathbb{Z}$ is, by the Chinese remainder theorem, isomorphic to $\prod_p \mathbb{Z}_p$. I ...