Questions related to the algebraic structure of algebraic integers

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0
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2answers
142 views

irreducible polynomial in $Z_p [x]$

Let $p$ be a prime integer. prove that the following are equivalent (a) $p$ is a prime in $Z[i]$ (b) $x^2 +1$ is irreducible in $Z_p [x]$ What I know is that if p is a prime in $Z[i]$, $p$ is ...
1
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2answers
100 views

Determination of the prime ideals lying over $2$ in a quadratic order

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, $D$ its discriminant. By this question, $1, \omega = \frac{(D + \sqrt D)}{2}$ is a basis of $R$ as a $\mathbb{Z}$-module. Let $x_1,\...
-4
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1answer
145 views

Determination of the prime ideals lying over an odd prime in a quadratic order

We need some notation before we state the problem. Let $K$ be a quadratic number field, $d$ its discriminant. Let $R$ be an order of $K$, i.e. a subring of $K$ which is a free $\mathbb{Z}$-module of ...
1
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2answers
179 views

Finding number of positive integral solutions of $x^4-y^4=3879108$

Find the number of positive integral solutions of $$x^4-y^4=3879108$$ $$3879108=36*277*389$$ I tried simplifying factors of $3879108$ to get terms in the form of $x^4-y^4$. However, I am unable to ...
-1
votes
1answer
445 views

Conductor of a quadratic order

We need some definitions to state the problem. Let $B$ be a commutative ring, $A$ its subring. We denote by $(A : B)$ the set $\{x \in B | xB \subset A\}$. $(A : B)$ is an ideal of $B$. It is ...
1
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0answers
65 views

Cyclotomic integers with complex norm $1$.

Let $R=\Bbb{Z}[\zeta_n]$. What are the elements in this ring with complex norm $1$? Are only $\zeta_n^i$ all such elements? What if $n$ is odd prime?
-1
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1answer
91 views

Condition for $a, b + \omega$ to be the canonical basis of an ideal of a quadratic order

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, $D$ its discriminant. By this question, $1, \omega = \frac{(D + \sqrt D)}{2}$ is a basis of $R$ as a $\mathbb{Z}$-module. Let $I$ be a ...
-4
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1answer
168 views

Canonical basis of an ideal of a quadratic order

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, $D$ its discriminant. I am interested in the ideal theory on $R$ because it is closely related to the theory of binary quadratic forms ...
3
votes
2answers
61 views

Isomorphism of $\mathcal{O}_K$-modules

I came across the following claim in K Conrad's notes: Let $L/K$ be a finite extension of number fields, For fractional ideals $\mathfrak{a}, \mathfrak{b}$, and $\mathfrak{c}$ of $\mathcal{O}_L$, with ...
3
votes
2answers
284 views

Cyclotomic Integers?

Let $K=\mathbb{Q}(\zeta_{p^\infty})$ be the extension of $\mathbb{Q}$ obtained by adjoining all $p-$power roots of unity. My question is : how to show that the the ring of integes of $K,$ $O_K$ is ...
0
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0answers
70 views

Smart way to compute decomposition of a prime

Let $f(x) = X^5 -4x +2$. I am wondering if there is a way to compute the decomposition of $(5)$ in $K = \mathbb{Q}[X]/(f(X))$ without computing the decomposition of $f$ in $\mathbb{F}_5$. What I'm ...
5
votes
1answer
224 views

Properties of squares in $\mathbb Q_p$

Let $\mathbb Q_p$ be the field of $p$-adic numbers. I know that for $p \neq 2$ an element $x=p^n u \in \mathbb Q_p^\times$ (with $n \in \mathbb Z$ and $u \in \mathbb Z_p^\times$) is a square if and ...
1
vote
2answers
64 views

Is it true that $(4+\sqrt{14})(4-\sqrt{14}) = (4+\sqrt{14})^2$ in $\mathbb{Q}(\sqrt{14})$?

Is it true that $(4+\sqrt{14})(4-\sqrt{14}) = (4+\sqrt{14})^2$ in $\mathbb{Q}(\sqrt{14})$? I am going through the solution of a problem I'm working on and this seems to be what they are saying. If its ...
3
votes
2answers
370 views

If $\alpha$ and $\beta$ are algebraic integers then any solution to $x^2+\alpha x + \beta = 0$ is also an algebraic integer.

An algebraic integer is a complex number that is a root of a monic polynomial with coefficients in $\mathbb{Z}$. Let $\alpha$ and $\beta$ be algebraic integers. Then any solution to $x^2+\alpha x ...
2
votes
0answers
80 views

Find all Integers ($ n$) such that $n\neq 6xy\pm x\pm y$

I am interested in proving that there exist an infinite number of positive integers ($n$) which are not of the form $$ n=6xy\pm x\pm y $$ for $x,y\in\Bbb Z^+$. [Note: The $\pm$ signs above are ...
0
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1answer
43 views

Why $\mathbb{Z}[\alpha]/23\mathbb{Z}[\alpha] \cong \mathbb{F}_{23}/(x^3-x-1)$ where $\alpha$ satisfies $\alpha^3=\alpha+1$?

This a step in my notes which I can't seem to understand clearly. Why $\mathbb{Z}[\alpha]/23\mathbb{Z}[\alpha] \cong \mathbb{F}_{23}/(x^3-x-1)$ where $\alpha$ satisfies $\alpha^3=\alpha+1$? I see ...
4
votes
2answers
2k views

Ideal generated by 3 and $1+\sqrt{-5}$ is not a principal ideal in the ring $\Bbb Z[\sqrt{-5}]$

Show that the ideal generated by 3 and $1+\sqrt{-5}$ is not a principal ideal in the ring $\Bbb Z[\sqrt{-5}]$. I fail to understand how can 3 and $1+\sqrt{-5}$ generate an ideal. $$(3,1+\sqrt{-5}) =...
1
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0answers
51 views

Examples of idelic Artin map

I do not know of any source on class field theory which explains the idelic Artin map through a couple of examples. Please provide me some.
6
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0answers
94 views

Isomorphism of algebras $E_v = \prod_{w\mid v} E_w$

I am currently reading the article Bayer–Fluckiger, Eva, B. Nivedita, and Raman Parimala. "Hasse principle for G–quadratic forms." Documenta Mathematica 18 (2013): 383-392. in which there is a ...
3
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0answers
106 views

special values of zeta function and L-functions

I was reading in some lectures notes about the Riemann zeta-function which takes on special values: $$\zeta(2) = \sum \frac{1}{n^2} = \frac{\pi^2}{6}$$ In fact, we can compute even values of the ...
24
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1answer
474 views

Intuition for Class Numbers

So I've been thinking about the analytic class number formula lately, and class numbers in general and I'm trying to develop a good intuition for them. My basic question, which may be too general/...
4
votes
1answer
298 views

Frey Curve as a Solution to FLT

I have read in many places that the Frey curve (if it existed) $y^2=x(x-A)(x+B)$ (or equivalently, $y^2=x(x-A)(x-C)$ corresponds to the solutions of $a^n+b^n=c^n$, where $A=a^n/c^n$ and $B=b^n/c^n$. ...
0
votes
1answer
154 views

On a certain basis of an order of a quadratic number field

Let $K$ be an algebraic number field of degree $n$. An order of $K$ is a subring $R$ of $K$ such that $R$ is a free $\mathbb{Z}$-module of rank $n$. Let $\alpha_1, \cdots, \alpha_n$ be a basis of $R$ ...
4
votes
1answer
89 views

perfect squares possible?

If we let a, b, c, d, and x be integers is it possible that $$x^2+a^2 = (x+1)^2 + b^2 = (x+2)^2 + c^2 = (x+3)^2 + d^2$$ My initial thought is no way! I tried expanding and simplifying, getting $$a^2 =...
5
votes
1answer
126 views

The ring of integers in $\mathbb{Q}[\zeta]$ is $\mathbb{Z}[\zeta]$

I am working on a proof in the lecture of Milne "Proposition 6.2 b)" but there is a step I don't get: We have an inclusion $\mathbb{Z}\hookrightarrow \mathcal{O}_K$ that induces the following ...
8
votes
1answer
170 views

Geometric intuition behind the Hasse principle

Let $f(X,Y) \in \mathbb{Q}[X,Y]$ be a quadratic polynomial. The Hasse-Minkowski theorem says that $f(X,Y) = 0$ has a solution $(x,y) \in \mathbb{Q}^2$ iff it has a solution in $\mathbb{R}^2$ and $\...
4
votes
2answers
255 views

Showing that the field generated by elements $e^{2\pi i/n} (n=1,2,…)$ is algebraic over $\mathbb{Q}$

Let $K$ be the field generated by the elements $e^{2\pi i/n} (n=1,2,...)$. Show that $K$ is an algebraic extension of $\mathbb{Q}$, but that $[K:\mathbb{Q}]$ is not finite. Suggestion: it may help to ...
12
votes
2answers
451 views

Are there Groups of Strictly Primes

Motivation Since Euclid's proof of the infinitude of the primes, the structure and properties of primes has always fascinated mathematicians. This lead to great work in their properties and ...
2
votes
1answer
229 views

On the prime in the ring of integers of infinite Galois extensions [duplicate]

I need help to solve the following exercise: Let $K$ be an infinite algebraic extension of $\mathbb Q$ and let $O_K$ denote the ring of algebraic integers in $K.$ If $\frak P$ is a prime ideal of $...
1
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1answer
164 views

Ramification of a Galois extension

I understand that an extension of number field $L/K$ is unramified if every non-zero prime ideal of $\mathcal{O}_K$ is unramified in $L$ (where a prime ideal $\mathfrak{p}$ of $\mathcal{O}_K$ is ...
18
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1answer
447 views

Class group and factorizations

There is a common characterization of the class group ${\rm Cl}(R)$ as a kind of measure of how badly factorization fails to be unique. The most obvious justification for this sentiment is that the ...
2
votes
2answers
164 views

Adelic topology on the group of ideles

The topology on $\mathbb{A}^\times$ is the subspace topology with respect to $\prod_v \mathbb{Q}_v^\times$ and a basis is given by the sets $$\prod_v\Omega_v$$ with $\Omega_v\subset\mathbb{Q}_v^\times$...
4
votes
1answer
383 views

Finite abelian unramified $p$-extension of a number field

Let $K$ be a number field. How many finite abelian unramified $p$-extensions of $K$ are there and what are their Galois groups? My feeling is, that every group $\mathbb{Z} / p^n \mathbb{Z}$ can occur ...
6
votes
2answers
240 views

infinite Algebraic extensions of $\mathbb{Q}$

I need help to solve the following exercise: If $K$ is an algebraic extension of $\mathbb{Q}$ (finite or infinite), then we let $O_{K}$ denote the ring of algebraic integers in $K.$ If $\frak{P}$ is a ...
2
votes
1answer
69 views

How is this homomorphism from Algebraic Number Theory surjective?

This is on page 72 of Cassels-Fröhlich's Algebraic Number Theory. Take $S$ to be the archimedian valuations of $k$, a finite extension of $\mathbb{Q}$, and define $J_S \subset V_k $ as the subgroup ...
4
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0answers
71 views

Solve $f(x)\mid g^2(x)+1$ in $\mathbb Z[x]$

We know that if $p\in \mathbb P$ and $p\equiv 1\bmod 4$ then we can find $t\in\mathbb Z$ such that $p\mid t^2+1.$ For what polynomial $f(x)\in \mathbb Z[x]$, we can find $g(x)\in \mathbb Z[x]$ ...
1
vote
2answers
91 views

Decomposition of an ideal as a product of two ideals

How to show $$5\mathbb{Z}[\sqrt[3]{2}] = (5, \sqrt[3]{2}+2)(5, (\sqrt[3]{2})^2+3\sqrt[3]{2}-1).$$ Firstly, I think that I can say that $$(5, \sqrt[3]{2}+2)(5, (\sqrt[3]{2})^2+3\sqrt[3]{2}-1)= (25,5(\...
2
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0answers
138 views

Let n be a nonzero integer and p an odd prime not dividing n.

$p/(x^2+ny^2)$ for $x$, $y$ relatively prime $\Leftrightarrow$ to $(\frac{-n}{p})=1$. I have proved the "$\Rightarrow$" part by using the fact that $y$ and $p$ must be relatively prime, which implies ...
3
votes
1answer
109 views

Discrete valuation on $p$-adic numbers

For the ring of $p$-adic integers $\mathbb Z_p$ let $\mu_n: \mathbb Z_p \to \mathbb Z / p^n \mathbb Z$ be the projection mapping. Consider $\mathbb Z / p^n \mathbb Z$ with the discrete topology. Is (...
6
votes
3answers
321 views

Overrings of Dedekind domains as localizations

I am taking an independent study where I organize and present weekly material on algebraic number theory to my professor and receive feedback. Next week I am going to cover some miscellaneous topics, ...
0
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1answer
84 views

work out the value of a - b from the identity $ax+18=2(x-b)$

How do I solve the following question? You are given the algebraic identity: $ax+18=2(x-b)$ Work out the values of $a-b$
2
votes
2answers
100 views

Is $\mathcal O_L$ an $\mathcal O_K$-lattice in $L$?

This is a basic question. Let $L/K$ be a finite extension of algebraic number fields and let $\mathcal O_L$ and $\mathcal O_K$ be their respective rings of integers. Is it true that $$K\otimes_{\...
15
votes
2answers
377 views

Are there applications of noncommutative geometry to number theory?

The marriage of algebraic geometry and number theory was celebrated in the twentieth century by the school of Grothendieck. As a consequence, number theory has been completely transformed. On the ...
3
votes
1answer
49 views

Augmenting «$\Bbb Z[x]$ f.g. $\Rightarrow x$ integral» for ${\frak p}[x]$

In KCd's blurb on ideal factorization, page 5: $\hskip 0.3in$ The situation is this: $K$ is a number field, ${\cal O}_K$ its ring of integers, ${\frak p}\triangleleft{\cal O}_K$ a prime ideal, $x\...
3
votes
2answers
211 views

Ramified prime in cyclotomic extension of a number field

Let $K$ be a number field, $n$ be a positive integer and $\zeta_n$ a primitve $n^{th}$ root of unity. How does one show that if a prime ideal $\mathfrak{p}$ of $K$ is ramified in $K(\zeta_n)$ then ...
7
votes
3answers
284 views

Integers of the form $x^2-ny^2$

Is there any algorithm to represent the given integers in the form $N=x^2-ny^2$, $n>1$, say for $n=2$. As we have for any prime, $p$, a prime $q$ can be written as $q=x^2+py^2$,whenever $q$ is ...
3
votes
2answers
289 views

What is meant by 'the completion of Z'?

In the first chapter of Algebraic Number Theory (lecture notes collected by Cassels-Fröhlich), page 28 has the following paragraph: "We suppose now that $k$ is a finite field of characteristic $p$ ...
2
votes
0answers
165 views

Symmetric function theorem and Galois Theory — How deep is the connection?

By symmetric function theorem in the title, the fundamental theorem of symmetric polynomials is meant: Any symmetric polynomial has a unique representation as a polynomial in the elementary symmetric ...
2
votes
1answer
557 views

On orders of a quadratic number field

Let $K$ be an algebraic number field of degree $n$. An order of $K$ is a subring $R$ of $K$ such that $R$ is a free $\mathbb{Z}$-module of rank $n$. Let $\alpha_1, \cdots, \alpha_n$ be a basis of $R$ ...
4
votes
1answer
330 views

What is Kronecker's Jugendtraum originally?

What was the exact statement of Kronecker's Jugendtraum according to Kronecker himself? Almost every new idea in algebraic number theory from Kronecker is cited as a progress towards Kronecker's ...