Questions related to the algebraic structure of algebraic integers

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1answer
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Basis for sum of all extensions of a completion: $\bigoplus_{w \mid v} \mathcal{O}_{w}$ over $\mathcal{O}_{v}$

I was going over notes from a class and it was stated (without proof) that if $\xi_{1}, \ldots, \xi_{n}$ is a basis of $K/k$, then for almost all places $v$, $\xi_{1}, \ldots, \xi_{n}$ is a basis for ...
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Mellin transform on $\mathbb{Z}[\omega]$

Let $\omega=\frac{-1+i\sqrt{3}}{2}$ be a complex cube root of unity. The Eisenstein integers $\mathbb{Z}[\omega]$ (a unique factorization domain) are of the forms $a+b\omega$ where $a$ and $b$ are ...
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1answer
33 views

Calculating zeta functions over a field

I am learning about zeta functions and have been trying the following example: Calculate the zata function of $x_0x_1-x_2x_3$ over $\mathbb{F}_p$. Does there exist an easy formula for calculating ...
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0answers
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Inertia Degree in Cyclotomic Extensions

Let $\zeta$ be a primitive $l$th root of unity, where $l$ is prime. If $p$ is another prime number, let $f$ be the order of $p$ in $U(\mathbb{Z}/l \mathbb{Z})$. Then in $\mathbb{Z}[\zeta]$, $p$ ...
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4answers
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$\mathfrak{a}_{1} + \dots + \mathfrak{a}_{n} = A \Rightarrow \mathfrak{a}_{1}^{r_{1}} + \dots + \mathfrak{a}_{n}^{r_{n}} = A$

I have to prove the following : Let $A$ be a commutative ring with unity and let $\mathfrak{a}_{i}$ be ideals in $A$. Assume that $\mathfrak{a}_{1} + \dots + \mathfrak{a}_{n} = A$. Let $r_{i}$ be ...
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0answers
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Find the ring of algebraic integers. [duplicate]

Find the ring of algebraic integers in $K=\mathbb Q(\sqrt[3]{2})$. So, I know that $K=\{a+b\sqrt[3]{2}+c\sqrt[3]{2}^2 \mid a,b,c \in \mathbb Q\}$. My professor has done very little on this topic. ...
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2answers
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Normalization of a variety

I'm currently in a number theory course and this question popped up. As I'm not super familiar with algebraic geometry, I was wondering if my reasoning is correct: Show that $\mathbb{C}[X,Y]/(Y^2 - ...
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1answer
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$\overline{\mathbb{Z}}$ is not a Dedekind domain.

I have to prove the following statement : Let $\overline{\mathbb{Z}}$ be the ring of all algebraic integers in (a fixed choice of) $\overline{\mathbb{Q}}$. Then $\overline{\mathbb{Z}}$ is not a ...
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2answers
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Question about Thm 1.4.4 in ANT Alaca/Williams

I am studying Introductory ANT by Alaca/Williams, p12, theorem 1.4.4: "Let $m$ be a nonsquare integer such that $\mathbb {Z}+\mathbb{Z}\sqrt {m}$ is a PID. Let $p $ be an odd prime for which the ...
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1answer
126 views

Some Galois theory

I have a question on field extensions, and I can't seem to find precise answers when browsing through online notes etc. Here it is: suppose $K$ and $k$ are fields with $k \leq K$ and $[K : k] = m$ ...
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0answers
21 views

Non-Galois number fields and complex embeddings

Let $K$ be a number field. $K$ is a normal extension of $\mathbb{Q}$ iff $\exists f(x)\in\mathbb{Q}[x]: K$ is the splitting field for $f(x)$. A field extension is Galois iff it is normal and ...
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1answer
57 views

Finding the Discriminant of $f(x)=x^n+ax+b$ Using Differentiation

Greetings fellow Mathematics enthusiasts. I was hoping someone could offer me some advice on proving the following statement about the discriminant of a polynomial with degree $n$. Let ...
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0answers
64 views

Generalized class group of $\mathbb Q(\sqrt{-5})$

I follow the notation of Georges Gras: Class Field Theory, some of which I recall for convenience; feel free to skip the following lines if you are familiar with the notation. Let $K$ be a number ...
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1answer
50 views

Find all positive integer pairs $(x,y)$ and $(u,v)$ with certain relations.

Is there exists any positive integer pairs $(x,y)$ and $(u,v)$ for which, the relations, $x^2+y^2=u^2+v^2$ and $x^3+y^3=u^3+v^3$ are satisfied simultaneously?
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$(X+4)=X^4+X^3+X^2+X+1 \pmod{5}$ by ramification of prime ideals

In Milne's algebraic number theory notes, on page 65, there is the following example: $X^4+X^3+X^2+X+1\equiv(X+4)^4\pmod{5}$. And Milne asks: Why is that obvious? This comes after discussion of ...
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+100

How does (21) factor into prime ideals in the ring $\mathbb{Z}[\sqrt{-5}]$?

The text of the exercise is the following: Show that $\mathbb{Z}[\sqrt{-5}]$ is a Dedekind domain, and that the identities $21 = (4+\sqrt{−5}) \cdot (4 − \sqrt{−5})$ and $21 = 3 · 7$ represent two ...
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2answers
46 views

Prove all elements of $A$ is algebraic over $C$, if all elements of $A$ are algebraic over $B$ and $B$ are algebraic over $C$

Let there be 3 fields $A$, $B$ and $C$. If all elements of $A$ are algebraic over $B$ and all elements of $B$ are algebraic over $C$, prove that this implies that all elements of $A$ is algebraic ...
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0answers
11 views

Prime containing ideal in number ring divides the index of ideal

I'm working through Peter Stevenhagen's notes on Algebraic Number Theory, and the third section starts: In order to factor an ideal $I$ in a number ring $R$ [into its primary composition $I = ...
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55 views

Name of a certain set

I want to know if there is any already-standard way to refer to the set described as follows. Take the set of all primes in $\mathbb{Z}$, call it $\mathbb{P}$. Take the set of all finite products of ...
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0answers
68 views

What are the “hidden” symmetries in Goldbach Conjecture?

What are the "hidden" symmetries in Goldbach Conjecture ? If Goldback conjecture is true, the basic instinct is that there must exist some "symmetries" which ensure (and lead) such properties. As we ...
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0answers
48 views

What are the missing gaps to prove Goldbach Conjecture?

When Andrew Wiles proved FLT, all he needed to do was to prove "semi-stable elliptic curve case" of Shimura-Taniyama conjecture. He did not need to start from scratch, he just needed to fill this ...
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1answer
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Ideals, Dedekind domain and $\mathbb{Z}[\sqrt{-3}]$

I have the ideal $\mathfrak{a} = (2, 1 + \sqrt{-3})$ in $\mathbb{Z}[\sqrt{-3}]$. I have to show that $\mathfrak{a} \neq (2)$ but $\mathfrak{a}^{2} = (2)\mathfrak{a}$ and then conclude that ideals do ...
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2answers
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6
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1answer
64 views

Generalizing the Big Omega function to Integral Domains

The $\Omega(n)$ function counts the total number of prime factors of $n$ counting multiplicity. Obviously, this definition extends to any Unique Factorization Domain. I have two follow up questions: ...
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1answer
35 views

Showing $\mathbb{Z}+\mathbb{Z}(\frac{1+\sqrt m}{2})$ is a Euclidean domain

Does anyone know an elementary proof for the following proposition? It is stated without proof in my textbook: Let $m$ be a negative squarefree ineteger with $m = 1 \pmod 4$. Then the integral domain ...
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1answer
145 views

Linear independence of fractional powers

Reflecting on this recent MSE question, I was led to the following conjecture : Let $A=\left\lbrace x^y \mid x,y\in{\mathbb Q}_+ \right\rbrace$. If $\alpha,\beta,\gamma\in A$ are pairwise $\mathbb ...
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1answer
63 views

Largest ideal of a local field on which a character is trivial

Let $K$ be a nondiscrete locally compact field. Then fixing a character $\chi$ on $K$, any character on $K$ can be written as $t \mapsto \chi(xt)$ for some $x \in K$. For $E \leq K$ a closed ...
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0answers
35 views

Group of Units in Cyclotomic Integers

I'm trying to show that for any $p$-th root of unity $\zeta$, where $p$ is an odd prime, we have $\mathbb{Z}[\zeta]^{\ast} = (\zeta)\mathbb{Z}[\zeta + \zeta^{-1}]^{\ast}$. Obviously the $(\zeta)$ ...
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1answer
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What was Lame's proof?

In 1847, Lame gave a false proof of Fermat's Last Theorem by assuming that $\mathbb{Z}[r]$ is a UFD where $r$ is a primitive $p$th root of unity. The best description I've found is in the book ...
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1answer
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$\mathbb{Q}(\sqrt{m}, \sqrt{n})$ : ring of integers, integral basis and discriminant

In the following document, http://people.math.carleton.ca/~williams/papers/pdf/033.pdf, I found three results about biquadratic fields and their ring of integers. It's the proof of the first theorem ...
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2answers
38 views

Prove that $\mathbb{Z}[\zeta_{p} + \zeta_{p}^{-1}]$ is the ring of integers of $\mathbb{Q}(\zeta_{p} + \zeta_{p}^{-1})$

I'm a bit at a loss about what I can say in this situation. Do I have to show that $\zeta_{p} + \zeta_{p}^{-1}$ form an integral basis ? If I do, I have no idea how to do it. If not, can I use the ...
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3answers
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How to multiply and reduce ideals in quadratic number ring.

I am studying quadratic number rings and I have a problem with multiplying and reducing ideals, for example: Let $w=\sqrt{-14}$. Let $a=(5+w,2+w)$, $b=(4+w,2-w)$ be ideals in $\mathbb Z[w]$. Now, ...
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0answers
20 views

Norm and trace in a number field

In the book's Algebraic number theroy Ian Stewart, exercise 12 chapter two. $K$ is a number field, $N_K,T_K$ is a norma and trace. 12: Give examples to show that for fixed $\alpha$, $N_K(\alpha)$ ...
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1answer
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Algebraic Integers and Irreducible Polynomials

Let $\alpha$ be an algebraic integer and let $f$ be a monic polynomial over $\mathbb{Z}$ of least degree having $\alpha$ as a root. Prove that $f$ is irreducible. I am having so many troubles with ...
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0answers
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The irreducibility of the polynomial $x^p\pm px-t$, where $t$ is an integer, $p$ is a prime number.

Let $f(x)=x^p+px+1$, where $p$ is an odd prime. Prove that $f(x)$ is irreducible. This is an exercise of a course (linear algebra, the first chapter focus on the polynomial rings), and the exercise ...
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Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ , $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$, what is the intersection?

Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ and $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$, What is the intersection $F_\infty\cap K_\infty$? Here $\zeta_{2^n}$ is a ...
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0answers
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Algebraic Integers

If $\alpha$ and $\beta$ are algebraic integers then show $\alpha + \beta$ and $\alpha \times \beta$ are both algebraic integers. I know that an algebraic integer is a root of some monic polynomial ...
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1answer
58 views

Are all transcendental numbers theoretically accessible?

I apologize if the title (and the body) of this question is worded incorrectly, but I have no real experience in (transcendental) number theory, so it's probably the best I can do. I've been thinking ...
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1answer
38 views

Sum of roots of unity an algebraic integer proof

Let S be the sum of a finite number of nth roots of unity (where n is fixed, and the sum is non-zero). How do I go about showing that S is an algebraic integer in the cyclotomic field of order n ?
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2answers
341 views

Does an infinite polynomial define algebraic numbers?

As the title says, does a polynomial with an infinite number of terms define algebraic numbers as roots? An algebraic number is defined as a solution to a polynomial with rational coefficients, but ...
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1answer
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Terminology — lying over something

I just came across reading something like this: 'Let $\phi\in \text{Gal}(L/K)$ lie above $Frob\in \text{Gal}(K^{un}/K)$.' Where $Frob$ is the Frobenius automorphism and $K^{un}$ is the maximal ...
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1answer
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Does there exist a finite set of polynomials which do not have roots over any prime field?

The polynomial $x^2 + 1$ has a root in $Z_p$ if and only if $p \not\equiv 3 \mod 4$, and the polynomial $x^2 + x + 1$ has a root in $Z_p$ if and only if $p \not\equiv 2 \mod 3$. So each of the ...
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2answers
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Are these proposed rules for the canonical factorization of algebraic integers complete?

In $\mathbb{Z}$, the rules are fairly well established, a few minor quibbles notwithstanding. But in, say, $\mathbb{Z}[\sqrt{7}]$, there are, as far as I can tell, no established rules. What I've seen ...
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2answers
75 views

Show that $\sqrt{-6}$ is irreducible in $\mathbb{Z}+\mathbb{Z}\sqrt{-6}$

Suppose not. Then there exists $\alpha,\beta\in\mathbb{Z}+\mathbb{Z}\sqrt{-6}$ such that $\sqrt{-6}=\alpha\beta\implies\alpha,\beta$ are not units. I'm not really sure where to go from here. Any ...
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1answer
58 views

Decomposition group of a prime ideal and root of polynomials

Let $f(x)$ be a monic irreducible polynomial with integer coefficient. Let $K$ be the splitting field of $f$ and $\alpha$ one of its roots. Let $p$ a prime number such that $p$ does not divide ...
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1answer
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$\sqrt {-6}$ is not prime in $\mathbb{Z}+\mathbb{Z}\sqrt {-6}$

Suppose $\sqrt{-6}|(a+b\sqrt{-6})(c+d\sqrt{-6})$. I need to show that $\sqrt{-6}$ does not divide $(a+b\sqrt{-6})$ and does not divide $(c+d\sqrt{-6})$. I thought you might arrive at some ...
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1answer
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which algebraic number theory book with answers to selected questions for self-study?

All: Can anyone recommend some easy to follow algebraic number theory books with answers (hints) to selected questions for self-study ? If a have no answers to questions, but if you know if some ...
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1answer
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Abelian Kummer Extension

A field extension of the form $\mathbb{Q}(\zeta_n, \sqrt[n]{\beta})$ where $\zeta_n$ is a primitive $n$th root of unity and $\beta \in \mathbb{Q}(\zeta)$ is called a Kummer extension. Even though ...
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2answers
34 views

Orderings of $\mathbb Q[\zeta]$

I want to apply an Theorem, but for that I need to know, how many orderings the totally real subfield of the $p$-th cyclotomic field $\mathbb{Q}[\zeta]$ has. I think possible answers are $1$ or ...
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Any general “formula” solutions for higher order polynomial equation?

We know that fifth (or higher) degree polynomial equation has no general solution formula using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of ...