Questions related to the algebraic structure of algebraic integers

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Norm of the product of two regular ideals of an order of an algebraic number field

Let $K$ be an algebraic number field of degree $n$. Let $\mathcal{O}_K$ be the ring of algebraic integers. An order of $K$ is a subring $R$ of $K$ such that $R$ is a free $\mathbb{Z}$-module of rank ...
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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 ...
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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 $a-b$
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Why does a Norm Form have rational coefficients?

[From Wolfgang Schmidt's "Diophantine Approximation and Diophantine Equations"] "Let $K$ be a number field with $[K:\mathbb{Q}] = d$ and $L(X)$ be a linear form, say $L(X) = L(x_1,\dots,x_n) = ...
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Proving that a ring has only finitely many prime ideals

Let $D$ be a Dedeking domain, $\mathfrak{i}$ a nonzero ideal of $D$ and let $B=D/\mathfrak{i}$ be the quotient ring. Then $B$ is a noetherian ring, and every prime ideal of $B$ is maximal. I have ...
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Cyclotomic euclidean number fields

I´m here because I want to repeat my question about the norm-euclidean algorithm in a particluar cyclotomic integer ring. Let $L=\mathbb{Q}(\zeta_{32})$ and $A=\mathbb{Z}[\zeta_{32}]$. On the page ...
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Norm of an ideal

Would someone attempt to give me a simple explanation of how to compute the norm of an ideal. I like the definition |$O/a$| but find it difficult to apply. Perhaps the "determinant version" lends ...
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The order of $P(K)/P^+(K)$ in a quadratic number field

Let $K/\mathbb{Q}$ be a quadratic extension. Let $P(K)$ be the group of principle fractional ideals of $\mathcal{O}_K$. Let $P^+(K)$ be the subgroup of principle fractional ideals with generator with ...
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Algebraic and Transcendental Numbers - Set Theory

Denote $\mathbb Q$$[x]$ = set of polynomials with coefficients $c_1$, $c_2$, $...$ ,$c_n$ in $\mathbb Q$. A number $a$ is algebraic if there exists a polynomial $f(x)$ in $\mathbb Q$[x] such that ...
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On a certain property of units in the cyclotomic number field of an odd prime order

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $\epsilon$ be a unit ...
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A special type of prime decompositions in a subfield of a cyclotomic field

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $G$ be the Galois ...
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64 views

All valuations equal one : unit?

Let $F$ be a global field with integers $o$, and let $x \in F$. Does $|x|_v =1$ for all non-archimedean valuations of $F$ imply that $x \in o^\times$.
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Ramification of local field

Let $K$ be finite field, $E=K((x))$ and $F=K((x^n))$, that is $F$ is field of formal laurent series in $x^n$. I know that $E\cong F$ (because you can consider $x\rightarrow x^n$ ) I want to prove if ...
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Number of points on a curve in a finite field

From Ireland and Rosen Number theory book(ch11.#11) Consider the curve $y^2=x^{3}-Dx$ over $\mathbb{F}_{p}$, where $D \not= 0$. Call this curve $C_{1}$. It can be shown that the substitution ...
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Extending an exponential valuation to the completion of a field

I was reading the section on Completions in Neukirch's Algebraic Number Theory. Neukirch uses the term multiplicative valuation for what other authors seem to call absolute value. He uses the term ...
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Reference for proof of “Dedekind's Criterion”?

It was mentioned to me briefly in passing about a criterion for rings of integers, referred to as Dedekind's Criterion. The Criterion essentially said that a ring $\mathbb{Z}[\omega]$ (for ...
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the minimum polynomial of a unit

Let $A$ be a dvr of characteristic zero. Let $B/A$ be a finite integral extension of $A$. Suppose that there exists a unit $x$ in $B$ such that $B=A[x]$. What can we say about the minimal polynomial ...
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Valuation of maximal real subfield of cyclotomic field

I'm stuck on the proof of Theorem 4.14 in Washington - Introduction to Cyclotomic Fields. We take a prime power $n = p^m$ and define $\pi = \zeta_n - 1$, $\zeta_n$ a primitive $n$-th root of unity. ...
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63 views

Generating Same Ideal Class

If I have two prime ideals $\mathfrak{p}$ and $\mathfrak{q}$ with $\mathfrak{p} = (a)\mathfrak{q}$ where $(a)$ is a principal fractional ideal (that is, we can have $a$ not necessarily in our ring of ...
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How to remove $1$ at position $x$ ( in base $B$) from a number represented in Base $10$

I was going through a solution on code chef in which we needed to remove a $1$ from a position say $x$ (in Base $B$) from a number in Base $10$ if the representation of that number in base $B$ had a ...
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In $(\mathbf Z/p^r \mathbf Z)^*$, finding an element with order $p-1$.

Let $p$ be an odd prime number. I want to prove that $(\mathbf Z/p^r \mathbf Z)^*$ is an cyclic group. I have known that $\overline {p-1} \in (\mathbf Z/p^r \mathbf Z)^*$ is of order $p^{r-1}$. Since ...
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Learning Algebraic Number theory

I am looking for a good reference to self study algebraic number theory, as no undergraduate course is given at the university. I've web-searched a lot of online notes and courses, and I don't seem to ...
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How do I compute the norm of a non-principal ideal of the ring of integers of a quadratic field without using ''large'' results

I am trying to compute the norm of the ideal $I=(7, 1+\sqrt{15}) \trianglelefteq \mathbb Z[\sqrt{15}],$ the ring of integers of $\mathbb Q[\sqrt{15}].$ I knew $I^2$ would be principal, as $I\bar ...
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Order of element in algebraic group

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
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Reducing an ideal to an ideal generated by fewer elements.

For $d=-31$, and $I=(2, 1/2 +\sqrt{-31}/2)$ I've been told that $I\cdot\overline{I}=(2)$ I've written $I\cdot\overline{I}= (4, 1-\sqrt{-31} , 1+\sqrt{-31}, 8) $ In what ways am I allowed to reduce ...
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Prime decomposition in $\mathbb Z[x]/(x^3-x^2+x+1)$

If $K$ is the unique number field of discriminant $-44$, K is isomorphic to the field generated over $\mathbb Q$ by a root of the polynomial $x^3-x^2+x+1$ with $\mathcal O_K=\mathbb ...
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Primitive element theorem for finite fields

Primitive element theorem for finite fields Can you explain $2$ points in the proof of the proposition below $\bullet$ First $\alpha$ is the root of the polynomial $T^{p^s}-T$, because $\mathbb ...
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If prime p doesn't divide the class number, then if I is an ideal of $O_K$, and $I ^{p}$ is principal, then I is principal

If a prime p doesn't divide the class number of a number field K, then if I is a non-zero ideal of $O_K$, and $I ^{p}$ is principal, then I is principal.
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Two questions concerning ideal factorization and norm

$\bullet$ In $\mathbb Z[\sqrt{-5}]$ why is $(2)=(2,1+\sqrt{-5})(2,1-\sqrt{-5})$ Actually both ideals on the RHS contain $(2)$, but also their product ? Can we just multiply RHS in the normal sense; ...
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Extension of Completions of Number Fields

On p. 116 of Milne's notes on Algebraic Number Theory, he gives the following construction. Let $K$ be a field with a valuation $|\cdot|$ (archimedean or discrete nonarchimedean), and let $L$ be a ...
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definition of Artin map

Let $L/K$ be an unramified Abelian extension. Then the Artin symbol $ \left ( (L/K) / \mathfrak p \right) $ is defined for all prime ideals $\mathfrak p$ of $\mathcal O_K$. (because in an Abelian ...
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Quadratic Number Fields

I am currently studying elementary Algebraic Number Theory and came across the following statement: Any Number Field $K$ such that $[K:\mathbb{Q}] = 2$ is equal to $\mathbb{Q}(\sqrt{d})$ for a unique ...
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Finding an integral basis for a lattice defined in terms of an equation modulo p

Let p be a prime number, u relatively prime to p, and $\Lambda := \lbrace (a, b) \in \mathbb{Z}^2 : b \equiv au$ (mod p)$\rbrace$. How then can I find an integral basis $v_1, v_2$ for $\Lambda$? I ...
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Writing a Gauss sum as a sum over divisors

Let $\chi$ be a Dirichlet character modulo $q$ induced by a primitive character $\chi^*$ modulo $d$ for some divisor $d$ of $q$. Let $n$ be a positive integer, and consider the generalised Gauss sum ...
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prime ideal of integral closure on the decomposition iff lies above the prime ideal of the ring

I'm having troubles proving the following proposition. In every reference I read, they mark this proposition as "clear" or "trivial", but I am unable to prove it. Some help? Let $A$ be a Dedekind ...
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Cauchy sequence in a valuation ring

From Janusz's book algebraic number fields, chapter 2. Let $K$ be a complete field with respect to a non-Archimedean absolute value $|\cdot|$. Let $R$ be its valuation ring with maximal ideal ...
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$\operatorname{lcm}(a,b) = c$ and $\gcd(a,b) = d$ => $\operatorname{lcm}(\frac{a}{d},\frac{b}{d}) = \frac{c}{d}$ in a Euclidean domain or PID

I know that in an integral domain $c=\operatorname{lcm}(a,b)$ if and only if $a\mid c, b\mid c$ and if there exists $c'$ such that $a\mid c', b\mid c'$ then this implies that $c\mid c'$. And ...
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Extension of a discrete valuation on a complete field

Let $K$ be a complete field w.r.t discrete absolute value $|\cdot|_K$, $\mathcal O_K=\{x\in K:|x|_K\leq 1\}$. $L$ is an extension field with $[L:K]< \infty$ and let $\mathcal O_L$ be the ...
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Proof algebraic number fields Janusz, implies that every dedekind Ring is principal?

The book is Algebraic number fields, The ring $R$ is a Dedekind ring and $\mathcal{U}$ is a ideal of $R$. Janusz The first four lemmas is completely clear to me, the While i can't understand the ...
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Field notation and degree of extension

Consider a field $\mathbb Q (\sqrt5, \sqrt7,\sqrt{35})$ as an extension over $\mathbb Q$. What is the degree of the extension? I am confused by the notation here. Does $\mathbb Q (\sqrt5, ...
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Can a non-UFD quadratic integer ring have some irreducible numbers that are actually prime?

And if so, is there an efficient way to identify such primes? For example, in $\mathbb{Z}[\sqrt{-5}]$, it's clear that $17$ is irreducible. Among its first fifty multiples ($34$ through $850$) I was ...
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Algebraic relations between trigonometric numbers

Given $n\in2\Bbb N$, what is precise algebraic relation between $cos\frac{\pi}{n-1}$,$cos\frac{\pi}{n+1}$? Both numbers are algebraic, which implies there should be an algebraic relation between ...
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Eigenvalue of multiplication in a number field.

Let $\mathbb{K}$ be an algebraic number field. $b \in \mathbb{K}$ defines a linear transformation $\phi(b)$ on $\mathbb{K}/\mathbb{Q}$ by multiplication - $\phi(b)(x):=bx$ for all $x\in\mathbb{K}$. ...
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Constructing idele from a rational number.

I am a novice to concept of idele ,despite the fact that I have gone through all its expositions in standard literature. Excusing my ignorance,suppose I take q=396000. Does it mean that the idele q= ...
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generator of number field inside a given number ring

If $R$ is a number ring (i.e., a subring of a number field) and $K$ its fraction field. Why can we always find an element $a\in R$ such that $K = \mathbb Q(\alpha)$? Probably I am missing an easy ...
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For $m$ distinct fields among $\mathbb{Q}(\theta_1),\ldots,\mathbb{Q}(\theta_n)$ prove that $m\mid n$ and each field occurs $n/m$ times

I'm having some trouble with this problem, and I wanted to know if someone could help me out. Let $K=\mathbb{Q}(\theta)$ be an algebraic number field of degree $n$. Let ...
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How to construct a polynomial from a radix-term?

A term only composed of the following operatings shall henceforth be called a radix term because I don't know how these terms are called. A radix term $t$ is either an integer or a sum of two radix ...
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Lattice with conditions

In Book's Algebraic number theory, I. Stewart page 142: Theorem 7.2: If $p$ is prime of the form 4k+1 then $p$ is sum of two squares. Proof: The multiplicative group $G$ of the field $\mathbb ...
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Field, algebraic element

1) Let $E/F$ an extension and let $\alpha,\beta\in E$ be algebraic elements over $F$. If $\alpha\neq 0$, prove that $\alpha+\beta$, $\alpha\beta$ and $\alpha^{-1}$ are all algebraic over $F$. 2) If ...
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Example of non fractional ideal

Let $R$ be an integral domain with fraction field $K$, and let $I$ be an $R$-submodule of $K$. We say that $I$ is a fractional ideal of $R$ if $rI\subset R$ for some nonzero $r \in R$. My question ...