Questions related to the algebraic structure of algebraic integers

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Generalisation of Fermat's Little Theorem in Function Fields

There is a well-known generalisation of Fermat's Little Theorem as for any $a\in\mathbb{Z}$ and $n\in\mathbb{N}$ we have $$\sum_{d\mid n} \mu(n/d) a^d\equiv 0\pmod n.$$ where $\mu$ is the Möbius ...
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What is Dirichlet class number formula for d when d is NOT a fundamental discriminant?

According to Wikipedia, Dirichlet published a proof of the class number formula for quadratic fields in 1839, but it was stated in the language of quadratic forms. Let d be a fundamental ...
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What are the equivalent statements of GRH using the Möbius or Liouville functions?

We all know that Riemann Hypothesis can be stated as properties of $\mu$ or $\lambda$, particularly in terms of the random behaviour of those functions with "square root" bounds. Are there similar ...
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Why the terminology “global fields” and “local fields”

Let $p$ be a prime number. A global field is defined as a finite extension of $\mathbb Q$ or $\mathbb F_p(t)$. On the other hand one can show that a local field (which is by definition a complete ...
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On the relative discriminant of a cyclic extension of an algebraic number field whose relative degree is a prime number

Let $K$ be a cyclic extension of an algebraic number field $k$ whose relative degree is a prime number $l$. Hasse wrote(see below) in his "Bericht" that the relative discriminant of $K/k$ is of the ...
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Is this ring a Dedekind domain?

Let $p(x)\in \mathbb{Z}[x]$ be a monic, irreducible polynomial in $\mathbb{Z}[x]$. For the field $K:=\mathbb{Q}[x]/(p(x))$, its ring of algebraic integers $\mathcal{O}_K$ is always a Dedekind domain ...
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Can class number $h(d)$ equal to zero for some $d$?

We know that $L(1, \chi)$ is related to the class number $h(d)$ with a constant. And this is one way that we can prove $L(1, \chi)$ not vanish on $s = 1$. What confused me is: we know that class ...
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Quantitative aspect of Chebotarev Density Theorem

I recently learned the Chebotarev Density Theorem for global fields. As far as I have seen, all applications of CDT seem to focused around proving some set of prime ideals (or places in function ...
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Is there an explicit formula for the expression $(a\mathbb{Z}+b) \cap (a'\mathbb{Z}+b'),$ not involving $\cap$?

Thinking of $\mathbb{Z}$ as a ring, the ideals of $\mathbb{Z}$ are precisely those subsets of the form $a\mathbb{Z}.$ Hence intersections of ideals can be computed by taking lowest common multiples. ...
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Discrete valuation fields and representation as power series

Let $(K,v)$ be a discrete valuation field ($v$ is surjective). Let $\mathcal O$ be the ring of integers of $v$ and moreover let $\mathfrak p$ be the unique maximal ideal of $\mathcal O$. Then we have ...
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eigenvalues of sum of matrices with algebraic integers eigenvalues

Let $A, B$ be two matrices such that they both have all eigenvalues in $\mathbb{A}$, the ring of algebraic integers. The question is: it is true that the matrix $A+B$ does have all of its eigenvalues ...
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Unique factorization theorem in algebraic number theory

Consider the set $S = a + b \sqrt {-6}$, where $a$ and $b$ are integers. Now, to prove that unique factorization theorem does not hold in set $S$, we can take the example as follows: $$ 10 = 2 \cdot ...
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p-adics $\mathbb{Q}_p$ is a field if and only if $p$ is a power of a prime

I want to show that the ring $\mathbb{Q}_p$ is a field for any prime $p$, so I want to show that every nonzero element has an inverse. I thought of the following argument, but I can't seem to locate ...
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Show that the number $z=\sqrt[3]{4}-2i$ is algebraic, that is satisfied a polynomial equation with integer coefficients.

Show that the number $z=\sqrt[3]{4}-2i$ is algebraic, that is satisfied a polynomial equation with integer coefficients. I thought I could use the Fundamental theorem of Algebra, but it seems to be ...
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What does $\log^3$ stand for in this paper by K. Győry?

In this paper from 1980 K. Győry proved an upper bound for the absolute value of the solutions to a given Thue-Mahler equation, but I don't understand what he meant with $\log^3$. Namely, consider a ...
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What are the dual polyhedra of the face-centered cubic lattice?

For a given lattice $L$ we could define the set of points closest to one point more than any other. $$ \{ x \} = \min_{\ell \in L} \|x - \ell \| \in \mathbb{R}^3$$ This generalizes the "fractional ...
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Importance of the compactness of idele group

Si $k$ is a number field and $J_k$ is the group of $V_k^*$ of invertible elements of the adele ring $V_k$ with the induced topology given by the morphism $V_k\to V_k\times V_K,\ x\mapsto(x,x^{-1})$. ...
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On the genus of a curve and its set of rational points.

The genus $g$ of a nonsingular curve $C$ of degree $n$ is defined as $g = \frac{1}{2}(n-1)(n-2)$. Let $C(Q)$ denote the set of rational points on $C$. By Faltings, we know that $C (Q) < \infty$ for ...
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A polynomial with integer coefficients with prescribed set of roots modulo infinitely many primes

Given an infinite set of distinct prime numbers $p_1, p_2,\ldots,p_n,\ldots$, and arbitrary integres $a_1, a_2,\ldots,a_n,\ldots$, is there a nonzero polynomial $f(x)$ with integer coefficients such ...
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Local reciprocity map applied to norm

This question concerns the local reciprocity homomorphism $r_L : L^\times \to G_L^\text{ab}$, where $L$ is a local field with absolute Galois group $G_L$. If $K\subset L$ is a subfield and $r_K$ is ...
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Monic polynomial terminology

If the constant term of a monic polynomial is one or negative one, is there a name for that special kind of monic polynomial?
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Ramification of primes in number fields in terms of valuations

Let $L/K$ be a number field extension. Is there a definition of ramification of primes(both infinite and finite) in terms of the valuations induced? This answer gives a definition for infinite ...
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Factorization of ideal in field $\mathbb{Q}(\sqrt[3]{2})$ and its normal closure

So far I've worked only with quadratic fields, and I'm not sure how to work with 3rd roots. I have ideal $(5)$ and need to factor it in $\mathbb{Q}(\sqrt[3]{2})$ and its normal closure. I know that ...
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$\mathbb{Q}(\sqrt[3]{17})$ has class number $1$

Let $\alpha:=\mathbb{Q}(\sqrt[3]{17})$ and $K:=\mathbb{Q}(\alpha)$. We know that $$\mathcal{O}_K=\left\{\frac{a+b\alpha+c\alpha^2}{3}:a\equiv c\equiv -b\pmod{3}\right\}.$$ I have to show that $K$ has ...
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Product of “Fake”-Galois Conjugates

My apologies if this question ends up being a duplicate; I did my best to search for an answer, but I have no idea what to call this stuff I'm working with, so I couldn't really find much. There is a ...
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What is the intersection of $\mathbb Z$ with the ideal generated by $1-\zeta_n$?

For example, 1-(-1) is in the ideal <2>, whenever $n$ is even. Suppose $R=\Bbb Z[\zeta_n]$, and (by the above), that $n$ is odd. We know 1+$\zeta_n$ can be multiplied by ...
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Rational points on $4x^5 + y^2 = z^2$

Does the title curve have any nonzero rational points ? I have to admit that i didn't find any significant insight to this problem.
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What are the pre-requisites required to understand Milnor's book on algebraic K- theory?

I want to understand Steinitz’ theorem on the structure of finitely generated modules over Dedekind domains. I also want to have some general awareness regarding what Algebraic K-theory is about. ...
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Square and cubic roots in $\mathbb Q(\sqrt n)$

Here is my question : Let $n$ a squarefree positive integer, $m \ge 2$ an integer and $a+b \sqrt n \in\mathbb Q (\sqrt n).$ What (sufficient or necessary) conditions should $a$ and $b$ satisfy so ...
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P-adic expansion of rational number

Maybe this is a silly question but I really can not see how to get a p-adic expansion of a rational number. I do know the case of for an integer but how can I extend to the rational number case. If we ...
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How are unramified extensions of number fields formed?

Every extension is formed by adjoining a root of a polynomial. E.g.: Totally ramified = root of Eisenstein polynomial. Unramified over a local field = root of cyclotomic polynomial. What about ...
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Units become powers when lifted to unramified extensions?

Suppose $k$ is an algebraic number field, and $K$ is an unramified extension. I know: non-units $p\in k$ cannot become a power in $K$, or else the ideal they generate would become ramified in ...
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Volume of a convex body

Let $x = (x_1, \ldots, x_n) \in \mathbb{R}^n$ and define the linear forms $L_i(x) = \sum_{j=1}^{n}a_{ij}x_j$ where $a_{ij} \in \mathbb{R}$. Define the domain $C$ by $$C : \{x \in \mathbb{R}^n : ...
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When is $\mathbb{Z}[\theta]$ a Dedekind domain for an algebraic number $\theta$?

The title says all. Suppose that $f\in \mathbb{Z}[t]$ is an irreducible polynomial (over $\mathbb{Q}$) and $\theta$ is a root of $f$. Can we determine when is $\mathbb{Z}[\theta]$ is a Dedekind ...
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Factorization of extension is injective

Let A be a Dedekind domain with field of fractions K, and let B be the integral closure of A in a finite separable extension L of K. Now I want to show the map from Id(A) to Id(B) is injective. I know ...
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Localising Dedekind domains

I'm wondering if the following is true: Let $A\subset B$ be two Dedekind domains with $B$ integral over $A$. Let $Q$ be a non-zero prime ideal in $B$ and $P=Q\cap A$. Then the localisation of $B$ ...
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Which ideal contains which? Or are they the same?

If I'm understanding this correctly, $13$ in $\mathcal{O}_{\mathbb{Q}(\sqrt{-23})}$ is irreducible but not prime. From A & W we see that $x^2 \equiv -23 \bmod 13$ has solutions and therefore ...
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How can I integrate this? (for calculate value of L-function )

I want to calculate the definite integral: $$ \int_{0}^{1} \frac{x+x^{3}+x^{7}+x^{9}-x^{11}-x^{13}-x^{17}-x^{19}}{x(1-x^{20})}dx. $$ Indeed, I already know that $\int_{0}^{1} ...
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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 ...
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Help to check a proof about local prime ideal being principal?

Let $K$ be a number field, call it's ring of integers $\mathcal O_K$ and take a - possibly nonprincipal - prime ideal $\mathfrak q$. I have shown that $\mathcal O_K$ is Noetherian integral domain and ...
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Matrix Algebra over Algebraically Closed Field

In Maclachlan and Reid's The Arithmetic of Hyperbolic 3-Manifolds, when proving that quaternion algebras are simple, they make use of the fact that $M_2(K)$, where $K$ is an algebraically closed ...
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algebraic conjugate

Let $\alpha, \beta$ be real roots of an irreducible polynomial over the field of rational numbers (i.e., $\alpha, \beta$ are algebraic conjugates). Is it possible that $\beta=\alpha^2$?
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$K|\mathbb{Q}_p$ un-ramified if and only if $(d_K)=(1)$: help with a passage

I need some help in th last passage of this proof: Suppose $K|\mathbb{Q}_p$ is un-ramified and of degree $n$. then $K=\mathbb{Q}_p(\alpha)$, where $\alpha$ can be taken to be an integral unit in ...
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Un-ramified extension of $\mathbb{Q}_p$. A clarification on the construction

I'm following the proof given in Koblitz's book which roughly speaking builds the un-ramified extension of degree $f$ of $\mathbb{Q}_p$ as $\mathbb{Q}_p(\alpha)$, where $\alpha$ is a root of the lift ...
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For what integer values of n is $\tan (\pi /n)$ an algebraic integer?

In http://oberlin.edu/faculty/jcalcut/arctan.pdf Calcut implies that this is true except when n is of the form $2{{p}^{k}}$for p an odd prime and k a natural number. He shows earlier that $\tan (\pi ...
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Compute Takagi group of the extension $\mathbb Q(i,\sqrt{-5})/\mathbb Q(\sqrt{-5})$

Given an extension $L/K$ of number fields we define the Takagi group as the subgroup $$T_{L/K} = N_{L/K} (D_L) \cdot H_K \subseteq D_K$$ where $N_{L/K}$ is the relative norm, $D_\bullet$ is the ...
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Dirichlet density for number fields $K$?

Let $K$ be a number field. Let $P$ be a subset of the set of nonzero prime ideals in $K$. For $\mathfrak{p} \in P$, let $N(\mathfrak{p})$ be its absolute norm, so $N(\mathfrak{p}) = ...
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Higher ramification groups of Galois extension of order $p^2$

Let $p\in \mathbb{Z}$ be a prime number and $K/\mathbb{Q}$ be a Galois extension of degree $p^2$ over $\mathbb{Q}$. Suppose that $P\subset \mathcal{O}_K$ is the only prime ramified over $p$. Let ...
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The group defined by Gauss's definition of composition of forms

In article 242 of Disquisitiones, Gauss investigates the properties of the direct composition of two forms of the same discriminant. In this case, he gives a "natural" choice for such a composition. ...
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Ideal theoretic proof of the first inequality of global class field theory

In the old days(namely in the 1920s), global class field theory was stated and proved without using $\mathfrak p$-adic fields. I am interested in their methods, but unfortunately they were written in ...