Questions related to the algebraic structure of algebraic integers

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Show that the set $\{ P \in E(\mathbb{Q})\ |\ h(P) \le M \}$ is finite, for any constant $M$.

Show that the set $\{ P \in E(\mathbb{Q})\ |\ h(P) \le M \}$ is finite, for any constant $M$. Here $h(P)$ is logarithmic height of $P$, that is, $h(P):=\log H(P)$ and $H(P)=H(x)$, for $P=(x,y) \in E(...
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Example of a number field with only one complex place

In a number theory textbook they are asking for a field $F$ which has only one complex place. Can $F = \mathbb{Q}(\sqrt{-2})$ ? Can $F$ be or arbitrary degree?
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50 views

Quotient ring of Gaussian integers $\mathbb{Z}[i]/(a+bi)$ when $a$ and $b$ are NOT coprime

The isomorphism $\mathbb{Z}[i]/(a+bi) \cong \Bbb Z/(a^2+b^2)\Bbb Z$ is well-known, when the integers $a$ and $b$ are coprime. But what happens when they are not coprime, say $(a,b)=d>1$? — For ...
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1answer
36 views

Subfield of a cyclotomic number field where a prime $p$ is inert

I am reading this paper by Adleman,Lenstra on finding Irreducible polynomial over Finite field. Here in Section VI(Proof of correctness of Algo B) I came across this argument: Let $q_i $ be a prime ...
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1answer
80 views

Fraction field of $p$-adic power series ring

Let $L$ be a finite extension of $\mathbf{Q}_p$. Write $$\mathcal{O}_{\mathcal{E}} = \left\{ f = \sum_{k \in \mathbf{Z}} a_kT^k \in \mathcal{O}_{L}[[T,T^{-1}]] \mid \lim_{k \to -\infty} a_k = 0\right\...
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2answers
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Algebraic integers of a cubic extension

Apparently this should be a straightforward / standard homework problem, but I'm having trouble figuring it out. Let $D$ be a square-free integer not divisible by $3$. Let $\theta = \sqrt[3]{D}$, $K =...
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1answer
73 views

Finding the ring of integers of $\Bbb Q(\sqrt[4]{2})$

I know$^{(1)}$ that the ring of integers of $K=\Bbb Q(\sqrt[4]{2})$ is $\Bbb Z[\sqrt[4]{2}]$ and I would like to prove it. A related question is this one, but it doesn't answer mine. I computed ...
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1answer
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A prime ideal $\mathfrak{p} \subset \mathcal{O}_K$ lies above/over $p$ if $\mathfrak{p}\cap \mathbb{Z} = p\mathbb{Z}$ [duplicate]

In the concrete case that $\mathcal{O}_K = \mathbb{Z}[i]$, and $\mathfrak{p} = (1+i)$, how to make sense of $\mathfrak{p}\cap \mathbb{Z}$? I want to know if (1+i) lies above/over 2.
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190 views

The ring of integers of the composite of two fields

Let $K,L$ be two number fields and let $KL$ denote the composite field (the smallest subfield of $\mathbb{C}$ containing both $K$ and $L$). Denote respectively by $R,S$ and $T$ the ring of algebraic ...
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On determining the ring of integers of a cubic number field generated by a root of $x^3-x+1$

I have the following question: Let $\alpha$ be a root of the polynomial $f(x) = x^3-x+1$, and let $K = \mathbb{Q}(\alpha)$. Show that $\mathcal{O}_{K} = \mathbb{Z}[\alpha]$. As I understand it, ...
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1answer
1k views

Ring of integers of a cubic number field

Let $K=\mathbb{Q}(a)$ where $a^3=d$ where $d\neq 0, \pm 1$ is a square free integer. Show that $\Delta (1, a, a^2)=-27d^2$. By calculating the traces of $\theta, a\theta, a^2\theta$ where $\theta=u+...
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3answers
155 views

Ring of integers in a cubic extension

Let $L=\mathbb{Q}[\alpha]$, with $\alpha^3=10$. How can be proved that $$\frac{\alpha^2+\alpha+1}{3}$$ is in $O_L$, the ring of integers of $L$?
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1answer
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Find an integral basis of $\mathbb{Q}(\alpha)$ where $\alpha^3-\alpha-4=0$

Let $K=\mathbb{Q}(\alpha)$ where $\alpha$ has minimal polynomial $X^3-X-4$. Find an integral basis for $K$. I have calculated the discriminant of the minimal polynomial is $-2^2 \times 107$, so the ...
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1answer
140 views

Ring of integers of $K=\Bbb Q[u]$ where $u=\sqrt[3]{p^2q}$

Let $p,q$ be distinct prime numbers $\ge 5$ such that $pq^2 \not\equiv 1\mod9$. Let $K=\Bbb Q[u]$ where $u=\sqrt[3]{p^2q}$, and $A$ be the ring of integers of $K$. I have shown that $u,v=pqu^{-1}\in A$...
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45 views

Determining when ring of integers is $\mathbb{Z}[\theta]$

Something which is not difficult to prove is that if $K$ is a number field generated by an integer $\theta$, then the ring of integers $\mathfrak{O}_K$ is generated over $\mathbb{Z}$ by $\theta$ and ...
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1answer
76 views

Algorithms for finding the ring of integers

In the book's Algebraic Number theory, Ian StewarT, Third edition (page 51-52), has the following propositions: Theorem 2.20: Let $G$ be an additive subgroup of $\mathfrak{O}_K$ of rank equal to the ...
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136 views

How to compute the integral closure of $\Bbb{Z}$ in $\mathbb Q(\sqrt[n]{p})$?

We have the definition of integral closure that all the integral elements of A in B. Could we just compute the integral closure of certain A in B. I am considering such a problem that given a prime p, ...
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Show that the ring of integers $A$ of the cubic field $\mathbb Q[x]$ with $x^3=2$ is principal.

Show that the ring of integers $A$ of the cubic field $K=\mathbb Q[x]$ with $x^3=2$ is principal. The hint given in the book is to majorize the discriminant of $A$ by $D(1,x,x^2)$ and then use the ...
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1answer
419 views

Finding the ring of integers of $\mathbb Q[\alpha]$ with $\alpha^5=2\alpha+2$.

I am stuck with problem 22, chapter 3 in Marcus' book Number Fields which says: Suppose $\alpha^5=2\alpha+2$. Prove that the ring of integers of $\mathbb Q[\alpha]$ is $\mathbb Z[\alpha]$. Prove ...
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1answer
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Example of field's normal closure that's not Abelian?

Suppose $K$ is a global field, $L/K$ is a field extension, and $M$ = normal closure of $L$ (over $K$). Is it possible that Gal($M/L$) is not Abelian? In all cases I know, $L$ is formed from $K$ by ...
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62 views

what is the value of $\zeta_{\mathbb{Q}(i)}(-1)$?

We've been told over and over $\boxed{\zeta(-1) = 1 + 2 +3 + 4 + \dots = - \frac{1}{2}}$ can be do the same over number fields? What should be the reasonable value for the zeta function $F = \mathbb{...
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Adeles for function fields

Usually the ring of adeles is defined for number fields: if $K$ is a number field the ring of adeles of $K$ is: $$\mathbb A_K:=\prod_{v}' K_v \;\;\;\;\;\;\;\;\;\;\;\;\;(\ast)$$ where $v$ ranges ...
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36 views

Completion and Algebraic Closure

Suppose we start with a valued field $K$ and we want to find a field extension of $K$ that is algebraically closed and complete. The usual process is: Consider the completion $\hat{K}$ of $K$, then ...
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1answer
61 views

Have I found an example of norm-Euclidean failure in $\mathbb Z [\sqrt{14}]$?

Based on the proof that $\mathcal O_{\mathbb Q (\sqrt{-19})}$ is not Euclidean because it lacks universal side divisors, I have convinced myself that $\mathbb Z [\sqrt{14}]$ is Euclidean because it ...
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1answer
40 views

Valuation of Index of polynomial with Newton Polygon

I read here (page 237) that the valuation of the index of a polynomial is equal to the number of integer points below its Newton polygon. I am confused how this makes sense--the cited paper (this) ...
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2answers
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Determining whether an element belongs to ring of integers

Consider a number field $K=\mathbb{Q}[\alpha]$ and we wish to show that some other element $\beta \in K$ belongs to the ring of integers $\mathcal{O}_K$. Is it enough to show that the norm and trace ...
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1answer
27 views

Discriminant of real cyclotomic field

I know following theorem (and its proof): Let $K\subset L \subset M$ be number fields, $[L:K] = n, [M:L]=m$, and let $\{\alpha_1,\ldots,\alpha_n\}$ and $\{\beta_1,\ldots,\beta_m\}$ be bases for $L$...
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1answer
46 views

What is meant by a number to be a root of unity?

I am proving first case of Fermat's last theorem for regular primes by following Marcus' book "Number Fields". I have to prove following statement: If $\varepsilon$ is a unit in $\mathbb{Z}[\omega]...
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2answers
73 views

$\mathbb Z$ basis of the module $\mathbb Z [\zeta]$

Given an $n$-th root of unity $\zeta$, consider the $\mathbb Z$-module $M := \mathbb Z[\zeta]$. Does this module have a special name? Does a basis exist for every $n$? And if so, is there an ...
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2answers
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Embeddings of pure cubic field in complex field

I know that the complex embeddings (purely real included) for quadratic field $\mathbb{Q}[\sqrt{m}]$ where $m$ is square free integer, are $a+b\sqrt{m} \mapsto a+b\sqrt{m}$ $a+b\sqrt{m} \mapsto a-b\...
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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}...
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1answer
28 views

What is the definition for totally ramified extension for a global field?

What is the definition for totally ramified extension for a global field? For local fields it means the maximal prime ideal generated from the uniformizer totally ramifies. But what is the definition ...
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1answer
94 views

Another real quadratic integer ring, Euclidean but not norm-Euclidean, with norm function needing only two adjustments?

I have come to know about $\mathcal O_K$ with $K = \mathbb Q(\sqrt{69})$. The norm function needs to be adjusted to absolute value, as is the case with other real rings, but it also needs to be ...
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factoring polynomials in ring of integers modulo powerful number

I am having trouble finding info on how to factor polynomials in ring of integers modulo powerful number. For example: $x^2 - 1$ in $\textbf Z_{8}$. I know by tinkering around that $(x - 1)(x + 1)$...
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Can a Mersenne number ever be a Carmichael number?

Can a Mersenne number ever be a Carmichael number? More specifically, can a composite number $m$ of the form $2^n-1$ ever pass the test: $a^{m-1} \equiv 1 \mod m$ for all intergers $a >1$ (Fermat'...
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Completion of abelian extension

I don't understand this step in a proof: Let $K/\mathbb{Q}$ be an abelian extension, $p$ a ramified prime and $K_p$ the completion at that prime. Then $K_p/\mathbb{Q}_p$ is abelian. Why does ...
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2answers
37 views

Ring of algebraic integers as lattice points in the complex plane

Let, $i=\sqrt{-1}$ and $\omega = e^{\frac{2\pi i}{3}}$. I know that we can represent the ring of integers $\mathbb{Z}[i]$ and $\mathbb{Z}[\omega]$ as square and triangular lattice on complex plane ...
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Exercise 1.10 from Silverman “The Arithmetic of Elliptic Curves ”

I am having trouble with Silverman's exercise 1.10(b). The converse of (a) is easy because there is no integer solution to the equation when $p \equiv 3$ mod $4$. However, this method does not work ...
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Prime divisors of the conductor of an order of an algebraic number field

Let $f(X) \in \mathbb{Z}[X]$ be a monic irreducible polynomial. Let $\theta$ be a root of $f(X)$. Let $A = \mathbb{Z}[\theta]$. Let $K$ be the field of fractions of $A$. Let $B$ be the ring of ...
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Does every coset in a ring of integers contain a totally positive element?

Let $K$ be a number field with ring of integers $\mathcal O_K$, let $\mathfrak m$ be an ideal in $\mathcal O_K$ and let $a \in \mathcal O_K$ such that $(a, \mathfrak m) = 1$. Does there necessarily ...
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If $f$ has more than one root in $K$, then $f$ splits and $K/k$ is Galois?

Let $f \in k[x]$ be an irreducible polynomial of prime degree $p$ such that $K \cong k[x]/f(x)$ is a separable extension. How do I see that if $f$ has more than one root in $K$, then $f$ splits and $K/...
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1answer
19 views

Normal closure of a number field and a possible quadratic field in it

While reading about prime decomposition in number fields, I came across following statement (stated as a fact): Let $K$ be a number field and $d= \text{disc}(\mathcal{O}_K)$, then the normal ...
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1answer
41 views

Splitting of primes in real cyclotomic field

The question is from Marcus' book, "Number Fields" (exercise 12, Chapter 4) Let $\omega= e^{\frac{2\pi i}{m}}$ and $p$ be a rational prime not dividing $m$. Then how does $p$ split in $\mathbb{...
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A few questions on the Gaussian integers

I have a few questions surrounding the Gaussian integers, which I hope can be answered together in one fell swoop. The Gaussian integers are defined as $\mathbb{Z}[i] = \{x + iy : x, y \in \mathbb{Z}...
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Prove that a specific ring of integers is not monogenic

I'm trying to prove that the ring of integers of $K=\mathbb{Q}(\sqrt7, \sqrt13)$ is not of the form $ \mathbb {Z}[a]$ for some $a$. Unfortunately I can not figure out where to start. I tried to ...
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1answer
28 views

Factorization of primes in normal closure of Quartic Field

Motivation for the question comes from Marcus' book on Number Fields (exercise 13, Chapter 4). Let $K= \mathbb{Q}[\sqrt[4]{m}, i]$ where $i=\sqrt{-1}$, $m\in \mathbb{Z}$ and $m$ is not a square. ...
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Find remainder when $f(x^{12})$ is divided by $f(x)$

What will be the remainder when $f(x^{12})$ is divided by $f(x)$ where : $$f(x) = x^5 + x^4 + x^3 + x^2 + x +1$$ I have already tried but found no idea how I can do this question.
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is the compositum of a family of algebraic extensions algebraic?

Let $F$ be a field contained inside another field $K$. Let $\Gamma$ be an indexing set (possibly infinite). Let $\lbrace E_i\rbrace_{i\in\Gamma}$ be a collection of algebraic extensions of $F$ ...
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2answers
93 views

Why consider ramification only over number fields?

Is there a reason why one looks at ramification of prime ideals only over (rings of integers of) number fields? There surely are many more situations where one has rings with prime ideals.
3
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Sum of squares using Gaussian integers

Using Gaussian integers $a+ib$ ($a,b\in\mathbb{Z}$), one can prove that a prime $p\in\mathbb{Z}$ is sum of two squares in $\mathbb{Z}$ if and only if $p\equiv 1\pmod 4$. Question: Using Gaussian ...