Questions related to the algebraic structure of algebraic integers

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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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$\mathbb{Q}(\sqrt{m}, \sqrt{n})$ : 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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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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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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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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Algebraic Integers and Irreducible Polynomials

Let alpha be an algebraic integer and let f be a monic polynomial over Z of least degree having alpha as a root. Prove that f is irreducible. I am having so many troubles with this question. I have ...
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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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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
55 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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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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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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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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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
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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
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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 ...
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1answer
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Infinite primes of a number field

Let $K$ be a number field. I know that to each real and to each complex conjugate pair of embeddings of $K$ there corresponds exactly one prime (equivalence class of absolute values) of $K$. How do I ...
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1answer
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Unramification of Ideals in Pure Cubic fields

I need some explanation for this .Let $K=\mathbb Q{\sqrt[3]{m}} $ be a pure cubic field with non square element $\alpha $ in $K$ such that ideal $(\alpha) $ is an ideal square in K. Let $ ...
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1answer
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If a ring has its field of fraction as algebraic number field $K$, would this ring be $O_K$?

Suppose that ring has its field of fraction as algebraic number field $K$. Would this ring then be $O_K$, ring of integers? Also, for $O_K$, would subring of $O_K$ be integrally closed?
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Number fields and algebraic integer

Suppose there is a (algebraic) number field $K$. Algebraic integer is a root of some monic polynomial with coefficients in $\mathbb{Z}$. The elements of $K$ are the root of some monic polynomial ...
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1answer
28 views

Integral closure as topological closure

For a commutative ring $A$ you can define the integral closure of $A$ as $$\overline{A}^{\operatorname{int}}:=\lbrace x\in \operatorname{Quot}(A)\mid x\text{ is integral over } A \rbrace.$$ Since this ...
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1answer
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Proving $n^q$ is algebraic when $n\in \mathbb N$ and $q\in \mathbb Q$.

Proving $n^q$ is algebraic when $n\in \mathbb N$ and $q\in \mathbb Q$. Definition: A number is called algebraic if it is the root of a polynomial: $P(x)=a_nx^n+...+a_1x+a_0$ My idea was to approach ...
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Motivating mathematics(particularly algebraic number theory) through historical problems.

Most mathematical textbooks start a subject by going backwards, historically. They will define the terms that were invented to solve a problem in their polished form and then use these definitions and ...
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Hilbert class field of cubic field

Let $K=\mathbb Q(\sqrt[3]7) $ be a pure cubic field with class number 3. I want know how to compute its Hilbert Class Field. I know that its degree of extension is 3. Thank You in advance.
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Any Computational Number Theory Book, include software programs for key steps of the proofs of major theorem?

All: Can anyone recommend some Computational Number Theory Books, which include software programs for key steps of the proofs of major theorem ? Some computational number theory books only include ...
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3answers
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Explain why the determinant of $A$ is the index of the subring?

Let $a$ be an algebraic number, whose minimal polynomial has integral coefficients. Let $K = \Bbb Q(a)$ be an algebraic number field. Let $\mathcal O_K$ be the ring of integers in this algebraic ...
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Proof of Hermite-Minkowski's Theorem

I want to prove the following theorem by Hermite and Minkowski: For any given discriminant there are at most finitely many number fields with this discriminant. A very helpful step is that if ...
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When are the coordinates of the intersection points of plane curves actually algebraic conjugates

Suppose $f(x,y)\in\mathbb{Z}[x,y]$ is an irreducible polynomial defining a plane curve. Say I want to find the intersection of this plane curve with the line defined by g(x,y)=y-ax+b. One way to do ...
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Can one characterise a global field geometrically?

A global field is either a a finite extension of the rationals a finite extensions of $F_q(t)$ Alternatively, the second is the function field of an algebraic curve over a finite field. Is there ...
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Show that i is an element of the p-adic integers if and only if p congruent to 1 mod 4

This exercise was given in a graduate course on Local Class Field Theory. We want to prove that $i\in \mathbb{Z}_p$ (the $p-$adic integers) if and only if $p\equiv 1 \mod 4$. For $\Rightarrow$, we ...
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Elements in ring of algebraic integers.

If $K$ is a number field and $\mathfrak{O}_K$ the ring of algebraic integers. Let $\mathfrak{p}$ a prime ideal, for each $\alpha\in{\mathfrak{O}_K}$ if ...
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Diophantine equations which are easier to solve using $\mathbb{Z}[i]$ compared to $\mathbb{Z}$

I wanted to know applications of arithmetic in $\mathbb{Z}[i]$ that helps in some problems of $\mathbb{Z}$. I found a wonderful set of notes by Keith Conrad. Now I want to read more on a similar ...
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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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Polynomial transformation of the roots of another irreducible polynomial.

Suppose I have some monic irreducible polynomial $g(x)$ in $\mathbb{Z}[x]$ with distinct roots $r_1,r_2,\dots,r_n$. Suppose $f(x)$ is some other polynomial, not necessarily irreducible. Is there ...
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1answer
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class number of pure cubic fields and elliptic curves

I want to find generators to Mordell Weil group of the Elliptic Curve $y^2=x^3−6321363052$ and class number of $\mathbb Q(\sqrt[3]{6321363052})$. Some suggestions such as algorithm or softwares will ...
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Hilbert Class Field for pure cubic fields

I am new to class field theory, I want to study Hilbert class field for pure cubic fields. Which is the good source? Thank you in advance.
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Shtukas?$\mbox{}$

Does there exist an exposition of the significance of shtukas for someone who is mathematically literate but is largelly ignorant of Drinfeld modules? This arises in the work of Peter Scholze among ...
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what is the most easy to read Algebraic Geometry book? [duplicate]

All: what is the most easy to read (most accessible) Algebraic Geometry book ? (If possible, I am looking for an introduction book, maybe for undergraduate, and maybe similar to A Friendly ...
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1answer
49 views

Why a particular ring of integers is not generated by a single element

It says here in the Sage documentation that the ring of integers in the number field obtained from $$f(x) = x^3 + x^2 - 2x + 8$$ is not generated by a single element. How would one go about showing ...
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1answer
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What are major algebraic number theory attempts, results and progressions toward Goldbach's Conjecture?

To my understanding, most progress toward Goldbach's Conjecture has been made in analytic number theory. Progress has often based on sieve, asymptotic estimation or other analytic methods. What are ...
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Salvaging an Algorithm which Finds the Discriminant of a Number Field

I am reading the book "Algebraic Number Theory and Fermat's Last Theorem" by I. Stewart and D. Tall (3rd edition) and stumbled over one of the problems: Let $K = \mathbf{Q}(\theta)$ where $\theta \in ...
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1answer
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probability that two randomly selected integers of an imaginary quadratic field of class number 1 are coprime

Given an imaginary quadratic field $\mathbb{Q}(\sqrt{-D})$, where $D$ is a Heegner number (1, 2, 3, 7, 11, 19, 43, 67, 163), what is the probability that two randomly selected elements of that fields' ...
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Smallest value taken by a quadratic polynomial in two variables.

Let $p$ be a degree $2$ polynomial with integer coefficients, say $$p(x,y) = Ax^2 + By^2 + Cxy + Dx + Ey + F.$$ I would like to find an algorithm which solves the following: Problem 1: Given ...
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1answer
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What is the relationship between the trace/norm of a quaternion and the definition in field theory?

I'm having some trouble figuring out the relationship between the trace/norm of a quaternion element and the definition of trace/norm in the extensions of vector spaces. According to my number theory ...