Questions related to the algebraic structure of algebraic integers

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1answer
68 views

Algebraic numbers and their closure

Are all of the roots/zeroes of a polynomial of finite degree with algebraic coefficients algebraic? How about for a generalization of a polynomial wherein the indefinite is exponentiated to an ...
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0answers
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The existence of Pisot numbers in any real number field

Wikipedia claims that, given a real algebraic number field $K$ of degree $n$, there is an algebraic integer $r \in K$ of degree $n$ such that $r>1$, but every conjugate of $r$ has modulus $<1$ ...
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Embed local Galois groups in global Galois group

Let $k$ be a global field, $p$ be a rational prime and let $S$ be a set of primes of $k$ with density $\delta(S) = 1$. Let $\mathfrak{p} \in S$ be a prime and denote by $k_\mathfrak{p}$ the completion ...
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1answer
31 views

Quadratic Reciprocity as a consequence of Eisenstein Reciprocity

I was recently looking at the wikipedia page on Eisenstein Reciprocity, which says it "extends Quadratic Reciprocity." However, though the two do seem to be related, I don't completely understand how ...
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23 views

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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0answers
46 views

Integral elements with predescribed properties in quaternion orders

In the course of doing some calculations I have found myself wanting to answer the following question: Let $D/\mathbb{Q}$ be a quaternion algebra ramified at a prime $p$ and at $\infty$ and let ...
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2answers
62 views

Where can i find resourses to study this algebraic number theory?

Where can i find material to study depper the farey fractions (continued fractions)? I triying to solve problems like these: 1.- Show that two consecutive convergent at least one of them satisfy: ...
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2answers
69 views

References for elliptic curves over schemes

As in the title, I want some references about theories for elliptic curves over rings(not fields) or over schemes. I heard that behaviours(?) of such elliptic curves are not as simple as elliptic ...
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1answer
89 views

Is there a “geometric” interpretation of inert primes?

I am currently learning about étale morphisms and how they behave a lot like covering maps. Now if I'm in the example of the ring of integers of a number field, it is possible to think of it as a ...
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0answers
26 views

$\Bbb A_K'$ is a one dimensional $\Bbb A_K$ module

Let $\Bbb A_K'$ be the dual to the group of adeles $\Bbb A_K$ of some field $K$. Then $\Bbb A_K'$ is an $\Bbb A_K$ module by the prescription $$a\cdot \Psi(x) \mapsto \Psi(ax)$$ But why is $\Bbb ...
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$\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 ...
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0answers
27 views

integral basis for an arbitrary cubic Galois field

I wonder where I can find some information (possible a book) about finding an integral basis for cubic Galois fields? I know that for pure cubic fields there exists a simple criterion according to ...
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1answer
47 views

Exact sequence out of commutative exact diagram

I'm trying to get grip on the following commutative exact diagram: I know where the maps come from and could verify the exactness and the other maps. (It is induced by the long exact sequence of ...
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1answer
44 views

Legendre Symbol and infinite descent method

Use infinite descent method to prove that for $p$ an odd prime: $$ \left(\frac{2}{p}\right)=-1 $$ if $$ p \equiv\pm 3\pmod 8 $$ Please I don't know how to connect two ideas. I'm sure any help ...
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0answers
43 views

Genus field = Hilbert Class Field (Cox exercise 6.15)

Prove that the genus field of an imaginary quadratic field of $K$ equals its Hilbert Class Field if and only if for primitive forms of discriminant $d_k$, there is only one class per genus. ...
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1answer
50 views

How to show that a valuation ring has a unique maximal ideal?

A subring $R$ of a fi eld $K$ is said to be a valuation ring of $K$ if for each $x$ $\in$ $K^{*}$ we have either $x$ $\in$ R or $x^{-1}$ $\in$ $R$. How can I show that the valuation ring has a unique ...
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2answers
113 views

A quotient $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain is principal (Neukirch exer 1.3.5)

The exercise states: The quotient ring $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain by an ideal $\mathfrak{a}\ne 0$ is a principal ideal domain. The proof by localization ...
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1answer
81 views

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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1answer
37 views

How to show that $2$, $1+\sqrt{-3}$, and $1-\sqrt{-3}$ are primes in $Z[\sqrt{-3}]$?

Seems I should use something related to norm map.
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Given a norm of an Eisenstein integer how do we find this integer?

Say we are given the norm of an Eisenstein integer $Nr(f)=7$. How do we actually find the integer? The norm for any Eisenstein integer is defined as ...
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0answers
77 views

Proving $\mathbb{Z}[\sqrt{-2}]$, $\mathbb{Z}[\sqrt{-1}]$, $\mathbb{Z}[\sqrt{2}]$, and $\mathbb{Z}[\sqrt{3}]$ are euclidean.

I have this short class note from my graduate number theory: ~~~~~~~~~~~~~~~~~~~~~~ THEOREM: Assume that |norm(x + y*sqrt(d))| < 1 for any two rational numbers x and y with $|x| \leq 1/2$ and ...
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2answers
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meaning of integrality

I've just started studying algebraic number theory, and found myself reluctantly admitting the definition of "integrality". As the definition says, an element is called "integral" if it is a root of a ...
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25 views

$Tr(\alpha_{i}^{n})=\alpha_{1}^{n}+…+\alpha_{d}^{n}$

I want to show $Tr(\alpha_{i}^{n})=\alpha_{1}^{n}+...+\alpha_{d}^{n}$ where $\alpha_{i}$ is algebraic integer of degree d and the $\alpha_{j}$ are it's conjugates. Is there any quick way to show it ...
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1answer
45 views

Integral Closure in an Unramified Extension is Generated by a Single Element

Let $R$ be a discrete valuation ring with quotient field $K$, and $L/K$ a finite separable extension which is unramified over $K$. Also suppose that $K$ is complete with respect to the valuation of ...
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2answers
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Checking whether $\sqrt{2}$ is contained in the ideal $(2,\sqrt{-5}+1)$

Let $L=\mathbb{Q}(\sqrt{-5},\sqrt{2})$ and $I=(2,\sqrt{-5}+1)$ the ideal in $\mathcal{O}_L$, the ring of algebraic integers in $L$, which is generated by $2$ and $\sqrt{-5}+1$ I want to show that ...
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1answer
79 views

solving $x^3-2y^3=1$ using cubic number field

I am trying to solve the diophantine equation $x^3-2y^3=1$ using $\mathbb{Q}(\sqrt[3]{2}).$ I've read this link: Solve $x^3 +1 = 2y^3$ The following is what i have tried: Finding all integer ...
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1answer
36 views

Using discriminants to find order of extension

Any hints on how to show $[G:H]^{2}=\frac{disc(H)}{disc(G)}$,where G,H are free abelian groups of rank n and $H\subset G\subset K$,where K is a number field? Alternative formulation, how to relate ...
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1answer
15 views

The basis for additive subgroup is discriminant-invariant

i.e. given bases $\{\beta_{i}\}$ and $\{\gamma_{i}\}$ for S additive subgroup of number field K (degree n over $\mathbb{Q}$), then $disc(\{\beta_{i}\})=disc(\{\gamma_{i}\})$. any hints? Is that ...
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1answer
25 views

Basis for $O_{Q[\alpha]}$ existence theorem reference

Theorem 13 from Marcus Number fields "This is an integral basis $1,\frac{f_{1}(\alpha)}{d_{1}},\frac{f_{2}(\alpha)}{d_{2}},...,\frac{f_{n-1}(\alpha)}{d_{n-1}}$ ,where $f_{i}$ monic and ...
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0answers
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maximal vs non-maximal order in an algebraic number field

I am trying to determine whether an order in a (cubic) number field is maximal or not. I have picked up two different fields. One has a power basis the other does not have it. 1) Let ...
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3answers
107 views

Textbook for graduate number theory

I am attending a graduate number theory, the professor did not assign any textbook. The materials are somewhere along the advanced/algebraic level such as Ring of Gaussian Integers, Quadratic Number ...
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1answer
12 views

Solution of a congruence equation in a PID

If D is a PID and $\ a,b,m\in $ D, then the equation: $ ax\equiv b \pmod{m} $ has solution $x \in $ D iff $b$ divides $(a,m)$ I have proved the left to right implication but I'm trying so hard ...
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0answers
44 views

Which remarkable properties does the Hilbert Class Field have?

Let $L$ be the Hilbert Class Field of $K$, then: $Gal(L/K) \cong CL(K)$ by Artin reciprocity. though being Galois is not transitive in general, we nonetheless have for $K/\mathbb{Q}$ and $L/K$ ...
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0answers
31 views

A homeomorphism $\overline{k} \otimes_k K \rightarrow \bigoplus K_j$

Let $k$ be a field with a non-archimedean absolute value $||$, $K/k$ a separable extension of degree $N$. Also let $K = k(\beta)$, and $\mu$ the minimal polynomial of $\beta$ over $k$. If ...
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7answers
155 views

Trying to get a bound on the tail of the series for $\zeta(2)$

$\frac{\pi^2}{6} = \zeta(2) = \sum_{k=1}^\infty \frac{1}{k^2}$ I hope we agree. Now how do I get a grip on the tail end $\sum_{k \geq N} \frac{1}{k^2}$ which is the tail end which goes to zero? I ...
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1answer
37 views

Ring of Integers of fields

Let $K\subset L$ be fields. Does it follow that $O_K\subset O_L$? It seems to me that this is certainly true but I can't be too sure. I know that since $K\subset L$, then $|L:K|\geq2$. How else can I ...
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1answer
43 views

Generators of $\Gamma(7)$, congruence subgroup of modular group

L.s., I try to do some calculations on the Klein Quarctic curve, but there is a basic thing I don't know how to compute. Let $\Gamma(7)$ denote the congruence subgroup of the modular group PSL(2, ...
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1answer
54 views

Hilbert modular forms and Hecke operators over Q

Let F be a totally real field. We know that we can define a Hecke operator $T_\mathfrak{m}$ on the space of Hilbert modular forms over $F$, say with some level structure, for any ideal $\mathfrak{m}$ ...
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1answer
48 views

Interpretation of $S$-ideal class group

I have a problem understanding the interpretation of the ideal class group in the case of restricted ramifiction. Let $k$ be a number field and $S$ a set of primes of $k$. Then $k_S$ denotes the ...
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0answers
55 views

How to solve diophantine equation $\frac{x^p-y^p}{x-y}=n$

$$\frac{x^p-y^p}{x-y}=n$$ whit $p$ a prime greater than or equal to $3$,for what value to $n$, it's solvable and how to solve,and whether $\frac{x^p-y^p}{x-y}=q_1$ $\frac{x^p-y^p}{x-y}=q_2$ is ...
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1answer
41 views

Question about the cyclotomic $\mathbb Z_p$-extension

Let $K$ be a number field and $K_{\infty}/K$ the cyclotomic $\mathbb Z_p$-extension of $K.$ My question is : How to prove that for any prime $\ell$ of $\mathbb Q$ distinct to $p$ does not decompose ...
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0answers
51 views

simple question regarding isomorphisms relating to ring of integers

I have a simple question about isomorphisms and ideals. Let $\mathcal O_F$ be the ring of integers in some quadratic number field $F=\mathbb{Q}(\sqrt d)$ and let $f(x)$ be the minimal polynomial of ...
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1answer
91 views

Volume of first cohomology of arithmetic complex

Let $K$ be a number field and consider the Arithmentic complex $\Gamma_{Ar}(1)^\bullet$ be defined by $$\begin{array} A\Bbb R^{r_1+r_2} & \stackrel{\Sigma}{\longrightarrow} & \Bbb R \\ ...
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0answers
44 views

algebraic integers in quadratic field

Can you give me a really complete proof tha in a quadratic fiield $Q[\sqrt{(d)}]$ the algebraic integers are: $\mathbb Z[\sqrt{(d)}]$ if $d\equiv2,3 \pmod4$ $\mathbb Z[(1+\sqrt{(d)})/2]$ if $d\equiv1 ...
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1answer
92 views

Algorithm to find solutions $(p,x,y)$ for the equation $p=x^2 + ny^2$.

As the classical book of David Cox argues, Assume the conditions are satisfied and $p$ can be represented as $x^2 + ny^2$. What would be a way to find solutions $(p,x,y)$ efficiently? Ideally, one ...
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0answers
51 views

Tricks to find the Hilbert Class field of a quadratic extension?

Let $L$ be the Hilbert Class Field of $K=\mathbb{Q}(\sqrt{-d})$. I already know, via Artin reciprocity, that $Gal(L/K) \cong CL(K)$. Another theorem (Cox 9.30) says that: $Gal(L/\mathbb{Q}) \cong ...
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1answer
33 views

Make Galois extension large enough to let Galois group act trivially on module

Let $K|k$ be a finite Galois extension of number fields, inside a given "maximal" (infinite) Galois extension $k_S$ of $k$. Let $G = Gal(k_S | K)$ denote the Galois group and let $A$ be a ...
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1answer
47 views

Ring of integers of $\mathbb{Q}(\omega_p,\omega_q)$

Let $p,q$ be distinct odd primes, and $\omega_p,\omega_q$ primitive $p$-th, $q$-th roots of unity. What is the ring of integers of $\mathbb{Q}(\omega_p,\omega_q)$? The numbers in the ring ...
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1answer
76 views

Interlude on Traces (and another interlude on how bad of a writer Frohlich is)

I'm trying to read through Frohlich's section of Algebraic Number Theory, but this guy really goes out of his way to make sure you don't understand anything. Frohlich is probably the guy Serre is ...
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2answers
122 views

Ring of integers for $\mathbb{Q}(\sqrt{23},\sqrt{3})$

What is the ring of integers for $\mathbb{Q}(\sqrt{23},\sqrt{3})$? So, these are numbers of the form $a+b\sqrt{3}+c\sqrt{23}+d\sqrt{69}$ where $a,b,c,d\in\mathbb{Q}$, and we want to find ones whose ...