Questions related to the algebraic structure of algebraic integers

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Finding a particular integral basis of the cyclotomic field

Let $\zeta_{39}$ be a primitive $39$th root of unity. How can I prove that all the conjugates of $\zeta_{39}$ form an integral basis of $\mathbb{Q}(\zeta_{39})$? This is from the paper "Cyclotomic ...
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1answer
71 views

Find primitive element of splitting field of $1 + x + x^2 - x^5$

As the title says, I need to find the primitive element of the splitting field of $1 + x + x^2 - x^5$ over $\mathbb{Q}$. Firstly, I would proceed by finding the roots as the splitting field has to ...
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Find extension of $\mathbb{Q}$ containing components of eigenvectors of a matrix

Given a matrix $\mathbf{A} \in \mathbb{Z}^{d \times d}$ I need to find an algebraic number $a$ of minimal degree, such that all eigenvalues and eigenvector's coordinates of $\mathbf{A}$ belong to the ...
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1answer
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Factorising ideals in $\mathbb{Z}[\sqrt{10}]$

I understand how to factorize ideals into prime ideals when they are of the form $(p)$, by Dedekind's Theorem, but I can't factorize ideals like $(4+\sqrt{10})$ in $\mathbb{Z}[\sqrt{10}]$. I can ...
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4answers
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Ordering the solutions to Pell's Equation

Let $S$ be the set of positive integer pairs $(x,y)$ such that $x^2 - d y^2 = -4$ or $x^2 - d y^2 = 4$, where $d$ is fixed as the discriminant of a real quadratic number field. I'm trying to show ...
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1answer
71 views

When does coprimality carry over to the base ring in an extension of Dedekind domains?

Let $A$ be a Dedekind domain. Let $K$ be the field of fractions of $A$ and $L$ is some finite field extension of $K$. Then let $B$ be the integral closure of $A$ in $L$. (Sorry I don't know how to ...
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2answers
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Solved Problems in Algebraic Number Theory

I'm not too sure whether this is the right place to ask this (and please correct me if it is not), but I'm currently studying a course in Algebraic Number Theory and would like to be pointed in the ...
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What are going to change of our view if $\pi+e$ is a rational? [closed]

It is well known that there's no conclusion now whether $\pi+e$ is a rational or not. Just for curiosity, what will happen if we know the answer?
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$\mathbb{Q}(\sqrt[p]{q}) \neq \mathbb{Q}(\sqrt[p]{r})$ for $p,q,r$ primes and $q \neq r$.

Let $p,q$ and $r$ be primes in $\mathbb{Z}$ with $q \neq r$. Let $\sqrt[p]{q}$ denote any root of $x^p-q$ and let $\sqrt[p]{r}$ denote any root of $x^p - r$. I need to prove that ...
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number theory of coefficients in an infinite sequence of polynomials

EDIT: equivalent formulation by Hurkyl in comments: if $n$ is odd and $p^\nu \parallel n$ and $n > 2k,$ then $$ p^{(\nu + 2 + 2 k - n)} \; | \; \sum_j \left( \begin{array}{c} n \\ 2j \end{array} ...
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37 views

Abelian extensions under inclusion, and their conductors

Suppose $K$ is a number field, and let $L$ and $L'$ be two abelian extensions of $K$, with conductors $C(L/K)=\mathcal{C}$ and $C(L'/K)=\mathcal{C}'$, respectively. Question: Is it true that the ...
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1answer
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Finding a primitive element for the Hilbert Class Field of $\mathbb{Q}(\sqrt{-14})$

I am trying to solve exercise 6.18 in the book of David Cox. I have tried to provide as much context as possible to make the situation as clear as possible for the reader. I have solved ...
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Etale cohomology and restricted direct product

[migrated to mathoverflow: http://mathoverflow.net/questions/161734/etale-cohomology-and-restricted-direct-product] $\newcommand{\h}{\operatorname{H}}$ Let $k$ be a global field, $A$ an abelian ...
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47 views

can the power of a complex number be arbitrary close to a number?

Given a complex number $z$ with $|z|=1$ and $z$ is not a root of unit, and a complex number $r$ with $|r|=1$, and a natural number $N>0$. Is it the case that for any $\epsilon>0$, there exists ...
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1answer
46 views

When is $\mathbb{Z}[\sqrt{d}]$ not an UFD (for $d>1$)?

I was wondering if there is a classification for this: For which $d$ is $D=\mathbb{Z}[\sqrt{d}]$ are UFD, with $d > 1$? For $d \equiv 1 $ (mod $ 4$), $D$ is not an UFD (proof here).
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1answer
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Extension of a discrete non-archimedean absolute value

Let $P$ be a prime ideal of a Dedekind domain $A$, $v$ an absolute value induced by $P$, and $B$ the integral closure of $A$ in a finite separable extension $E$ of $A$'s quotient field $K$. If $w$ is ...
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Majoration of the trace in local field

Let $p$ be a prime number. Let $L/K/\mathbb{Q}_p$ be finite extensions where $L$ has relative ramification degree $e$ over $K$. When I extend the $p$-adic absolute valute $|\cdot|$ to $L$ and $K$ I ...
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1answer
49 views

Sum of Three squares

Considering $z=a+bi+cj$ ($a,b,c\in\mathbb{Z}$) and $w=d+ei+fj$ ($d,e,f\in\mathbb{Z}$) and the property of complex numbers that $|zw|=|z||w|$. If the rule of multiplication $zw$ is defined such that ...
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Show that $x^2 + x + 12 = 3y^5$ has no integer solutions.

Show that $x^2 + x + 12 = 3y^5$ has no integer solutions. Use the fact that the class group of $K$ is cyclic of order 5, where $K=\mathbb{Q}[\alpha]$ and $\alpha$ is the root of $x^2-x+12$. We get ...
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1answer
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Dedekind's example of a cubic field $K$ for which $O_K$ does not have the form $\mathbb Z[\alpha]$

Let $\alpha$ be a root of $f(X)=X^3+X^2-2X+8$ and $\beta = \frac{4}{\alpha}$. It can be shown that $O_K=\mathbb{Z}[\alpha, \beta]$. How does one then establish the following ring isomorphism ...
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What primes can ramify and decompose in $k(\mu_{p^m}) \mid k$?

Let $k$ be a number field and $p$ be a rational prime. Then consider the extension $k(\mu_{p^m}) \mid k$ of adjoining all roots of unity of degree $p^m$ to $k$. Assuming that $\mu_{p^m} \cap k$ is ...
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1answer
35 views

Basis for an ideal in $\Bbb{Q}(\sqrt[3]{m})$

Let $m$ be a cube-free integer, so that $K=\Bbb{Q}(\sqrt[3]{m})$ is a cubic extension of the rationals. Then the ring of integers in $K$ is a free $\Bbb{Z}$-module of rank $3$, with a basis of the ...
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Prove that $A^B=\Bbb{Z}(\sqrt{2})$

$A^B =\{\,b \in B : \text{$b$ is integral over $A$}\,\}$ Let $A=\Bbb{Z}$ and $B=\Bbb{Q}[\sqrt{2}]$ . Prove that $A^B=\Bbb{Z}[\sqrt{2}]$ Thank you for your helping. i am sorry for my poor English ...
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Which lattices are ideals of a number field?

Let $K$ be a number field, then its ring of integers $\mathcal{O}_K$ in the Minkowski space of $K$ is a lattice $\Lambda$. Is there some geometric descrpition/intuition that describes sublattices of ...
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Are these cubic rings the same?

Consider the pure cubic field $K=\mathbb{Q}(\sqrt[3]{10})$ then as $10\equiv 1 \pmod 9$ then the integral basis for $K$ is of the form $\{1,\sqrt[3]{10},\frac{1+\sqrt[3]{10}+\sqrt[3]{10^2}}{3}\}$. And ...
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Factorization Patterns for Ideals

Let $K/\mathbb{Q}$ be a Galois Number field. Let $p$ be an unramified rational prime. In this extension, for any $P,Q | p\mathcal{O}_K$ then the relative degrees $f(P) := [\mathcal{O}_K/P : ...
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Mapping of inertia group in local class field theory

Let $p$ be a rational prime, $K$ be a local field, $K(p) \mid K$ be the maximal $p$-extension of $K$ inside a given separable closure. Now let $I(K(p) \mid K)$ be the inertia group of $K(p) \mid K$ ...
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Number field theoretical analogue of Riemann's Existence Theorem

This question is related to Riemann's Existence Theorem as cited in Neukirch, Schmidt et. al.: Cohomology of Number fields (10.5.1). Let $k$ be a number field, $T \supseteq S \subseteq S_p \cup ...
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1answer
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Precise Error Term in Chebotarev's Theorem

Let $K/\mathbb{Q}$ be a Galois Number Field with Galois group $G$ and discriminant $\Delta_K$. Chebotarev's theorem states that the number of (unramified) rational primes with Frobenius conjugacy ...
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1answer
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$p$-part of cyclotomic character

Let $K$ be a number field, $\bar K$ a separable closure and $p$ a rational prime and assume that $p \not= char(K)$. Consider the extension $$K(\mu_{p^\infty}) \mid K$$ which is obtained by adjoining ...
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1answer
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Scalar extension of simple algebras

I have a question regarding quaternion algebras. Let $K$ be a number field and $L|K$ a quadratic field extension. Let $M|K$ be a field extension such that $M\otimes_K L$ is not a field, i.e. ...
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49 views

Infinite primes and notation

While reading a book about algebraic number theory, the symbol for a rational prime $p$ $$p^\infty$$ often occurs and I was wondering, what the exact definition of this is. Also, what is the ...
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1answer
20 views

Formula for Hilbert symbol for primes lying above 2

Let $K$ be any number field, and let $\mathfrak{p}$ be a prime lying over 2. Is there a formula for computing the Hilbert symbol $(a,b)_\mathfrak{p}$? I know the formula when $\mathfrak{p}$ lies above ...
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1answer
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Number of field extensions of $\mathbb{Q}_p$

If I know the index $(\mathbb{Q}_p^{\times} : (\mathbb{Q}_p^{\times})^n)$ for some $n \in \mathbb{N}$, is it possible to know how many field extensions of $\mathbb{Q}_p$ of degree $n$ there are? This ...
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$p$ is an odd prime of the form $p=x^2+2y^2$ iff $p\equiv_8$ $1$ or $3$ [duplicate]

How would I prove the following: Show that an odd prime $p$ can be written on the form $p=x^2+2y^2$ for some $x,y\in\mathbb Z$ iff $p\equiv_8 1, 3$. Hint: use the quadratic reciprocity and the ...
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Do conjugate fields have the same maximal order?

Suppose that $\alpha,\beta\in K$ are conjugates elements, i.e. zeros of an irreducible polynomial over $\mathbb{Q}$. Then we know that the fields $K_{1}=\mathbb{Q}(\alpha)$ and ...
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On number of different factorizations over integers of a number field

Let $K$ be a finite field extension of the rational numbers and let $\mathcal{O}_K$ denote its ring of integers. If a rational integer $n$ factors into two distinct ways into irreducible elements in ...
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1answer
60 views

Completion of the unit group of a local field

Let $K$ be a number field and $\mathfrak{p}$ a finite prime of $K$. Denote the unit group of the ring of integers of the local field $K_\mathfrak{p}$ (i.e. the completion of $K$ via $\mathfrak{p}$) by ...
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Cyclotomic Character

I have a couple of questions concerning the cyclotomic character. For the moment I know very little about the mod $\ell$ cyclotomic character, namely that ...
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How to determine the density of the set of completely splitting primes for a finite extension?

In reply of sea turtles comment in this thread Let $k$ be a number field and $K \mid k$ a finite Galois extension. What is the density of the set of completely splitting primes of $k$? As sea turtle ...
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Embedding into $p$-adic complex numbers

As I'm reading notes about the Leopoldt conjecture, the following question came to my mind: Let $\mathbb{C}_p$ be the $p$-adic complex numbers, i.e. the completion of the algebraic closure of the ...
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Particular determinant made of powers of algebraic numbers is nonzero?

Let $P$ be a degree-two polynomial, with roots $\alpha,\beta$. Is there a simple condition on $P$ (or on $\alpha,\beta$), equivalent to the following : $$ ...
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1answer
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suggestion about elliptic curves

I have read a little bit of number theory and covered upto Kummer's proof of Fermat's Last Theorem for regular primes. I am familiar with the concepts like disciminant, class number. Could anyone tell ...
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108 views

If $\alpha$ is an algebraic number, then prove that $\frac{2}{3}\alpha$ is also algebraic.

If $\alpha$ is an algebraic number, then prove that $\frac{2}{3}\alpha$ is also algebraic. Note: $\alpha \in \mathbb{R}$ I know that if $\alpha$ is algebraic, then there exists some $f(x) \in ...
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Congruences or logs

How do you know the pairs of integers $x,y$ such that $$y=\frac{ln(p(x))}{log(k)}$$ is true, where $p(x)$ is any Diophantic equation and $k$ any Natural number?
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Density of set of splitting primes

Let $K$ be a number field and let $S$ be a set of primes of $K$ containing the set of archimedian primes $S_\infty$. Suppose, $S$ has Dirichlet density $\delta(S) = 1$. Then the claim is that the ...
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1answer
48 views

Questions about the square root of $a$ in $\mathbb{F}_p$.

How to prove that there is a square root of $3$ in $\mathbb{F}_p$ if and only if $p \equiv 1 $ or $11 \pmod {12}$? Thank you very much.
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The unramified quadratic extension of $\mathbb{Q}_2$

I know that there are $7$ field extensions of $\mathbb{Q}_2$ of degree $2$ (this follows from Hensel's lemma) and I think these are $$\mathbb{Q}_2(\sqrt{2}), \mathbb{Q}_2(\sqrt{3}), ...
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Definition of a splitting field of a finite group

This is a basic question from the journal 'Mathematische Zeitschrift' 208 (1991) page 243. Let $K/F$ be a finite Galois extension of number fields and $G={\rm Gal}(K/F)$. Also let $L$ be any number ...
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1answer
56 views

Nonconstructible Algebraic Numbers

What is an example of an algebraic number that is not Euclideanly constructible (meaning that no series of basic arithmetic operations or the taking of roots can produce the number)?