Questions related to the algebraic structure of algebraic integers

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1answer
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Group $U$ of $p$-adic units is inverse limit of $U/U_{n}$

In Serre's famous Course in Arithmetic, there is a somewhat unexplained claim: Let $U=\mathbb{Z}_{p}^{\times}$ be the group of $p$-adic units. For every $n\geq 1$, put ...
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1answer
21 views

Using Dedekind's prime ideal factorisation theorem

I've been going over past papers for algebraic number theory and came across this question which has given me some trouble: Given a number field $K =\mathbb{Q}(\sqrt{-d})$ where $ d\equiv 1 \mod 4$ ...
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1answer
37 views

Ramified primes in radical extension of number fields

Let $ K $ be a number field, $ n\ge2$ be a positive integer and $a \in K^*$. How does one show in the simplest possible way that a prime ideal $\mathfrak {p}$ of $ K $ not dividing $ n$ is ...
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0answers
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What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$?

Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$. If $L$ is a number field ...
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0answers
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Problem when computing ideal class group

When computing the ideal class group of a quadratic extension $\mathbb{Q}[\alpha]$ after we have decomposed all rational primes smaller than the Minkowski bound into generating prime ideals ...
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26 views

Neukirch ANT - Reciprocity map doesn't depend on primes

This is a quick question. Unfortunately, Neukirch does CFT in such a unique way that it is not very easy to explain all the notation and concepts he defined here, so I expect someone who read the book ...
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22 views

Computing $n^{\text{th}}$ root of a positive integer to arbitrary precision using integer arithmetic

There are various questions on this forum that appear similar, but my question pertains to writing code that can compute the $n^{\text{th}}$ root of a number $a$ correct to $p$ decimal places, where ...
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3answers
248 views

Importance of continuity of Galois representations

So for a one dimensional Galois representation $\rho: G_{\Bbb Q} \to \mathbb C^{\times}$, I know that it must factor through the abelianization of $G_{\Bbb Q}$, which by the Kronecker-Weber theorem is ...
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1answer
130 views

Ramification in the ring of all algebraic integers

If $F$ is a finite extension of $\mathbb{Q}$ then its of integers $R$ is a Dedekind domain, and has unique factorization of ideals into powers of prime ideals. For each prime number $\ell$, you can ...
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1answer
33 views

Isomorphism of completions of number fields

Let $K$ and $L$ be number fields, $v$ a place of $K$ (either archimedean or non-archimedean) and $\theta:K\simeq L$ a ring isomorphism. I am trying to show that $\theta$ induces an isomorphism ...
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2answers
302 views

Infinitely many primes in the ring of integers

Let $K$ be a number field such that $\mathcal{O}_K= \mathbb{Z}[\alpha]$ for some $\alpha$ algebraic integer. Prove that there are infinitely many primes $\mathcal{P} \subset \mathcal{O}_K$, such ...
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1answer
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formal derivative algebraic [closed]

Let $q$ be a prime power, $\mathbb{F}_q$ the field with $q$ elements and $f \in \mathbb{C}_\infty$ be of the form $f = \prod_{i=1}^\infty f_i$ with $f_i \in \mathbb{F}_q(X)$ (here $\mathbb{C}_\infty$ ...
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2answers
31 views

Completeness proofs for the solutions of Diophantine Equations

In general, what are the strategies for showing the completeness of a solution set for Diophantine Equations? For example, take the $\textit{Pell-type}$ equation $x^2 - dy^2 = a$. Say, you have a set ...
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1answer
44 views

Significance of the Riemann hypothesis to algebraic number theory?

Of course, the truth of the Riemann hypothesis is a central question in analytic number theory. Does its truth/falsehood have important consequences in purely algebraic number theory as well? ...
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Smallest Subset of $\mathbb{R}_{>0}$ Closed under Typical Operations

Let $S$ denote the smallest subset of $\mathbb{R}_{>0}$ which includes $1$, and is closed under addition, multiplication, reciprocation, and the function $x,y \in \mathbb{R}_{>0} \mapsto x^y.$ ...
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1answer
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A question about the definition of $p$-adic pseudo-measure.

Let $\mathfrak B$ be a profinite abelian group and let $\Lambda(\mathfrak B)$ be defined as the inverse limit $\varprojlim \mathbb Z_p[\mathfrak B/ \mathcal H]$ where the inverse limit is taken with ...
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1answer
436 views

An application of the Weak Approximation theorem - Artin-Whaples Approximation Theorem

Let us recall the weak approximation theorem from Valuation theory in Algebraic Number Theory. Let $K$ be a field, and let $|\cdot|_{1},\dots, |\cdot|_n$ be pairwise non-equivalent nontrivial ...
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0answers
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Adelic definition of “canonical divisor”

For a function field over a curve $F/K$, some book define the canonical divisor as the divisor of a map $\omega:\mathscr{A}_{F}\rightarrow K$ (where $\mathscr{A}_{F}$ is the pre-adele, ie. adele but ...
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0answers
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Does a place $v$ of a number field $K$ ramify in $L/K$ iff $v\mid d_L$?

Let $L/K$ be an extension of number fields and $v$ be a prime (an equivalence class of valuations) of $K$ and $d_L$ the absolute discriminant of $L$. I know that a rational prime $p$ in $\mathbb Q$ ...
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1answer
15 views

Splitting of a prime in the compositum of two fields [duplicate]

Let $L$ and $M$ be two finite extensions of $\mathbb{Q}$ and let $LM$ denote their compositum. Suppose that $p$ is a rational prime that splits completely in $L$ and $M$. How can I show that $p$ ...
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1answer
60 views

In $\mathbb{Z}[\omega]$, if $a^3+b^3+c^3=0$ then $1-\omega$ divides at least one of $a,b,c$

This is problem 3.26 (self-study) in "Ireland and Rosen" If $a,b,c \in \mathbb{Z}[\omega]$ and none are equal to zero, and $a^3 + b^3 +c^3 = 0$ , show at least one of $a,b,c$ is divisible by ...
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1answer
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Computing prime factorization of ideals?

I want to compute the prime factorizations of the ideals $\langle 4\sqrt{-14}\rangle$, $\langle 6\sqrt{-6} \rangle$ and $\langle 4\sqrt{-5} \rangle$ in the ring of algebraic integers of ...
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1answer
250 views

Connection between ramification in number fields and Clifford theory

Consider algebraic number fields $\mathbb{Q} \subseteq K \subseteq L$ with rings of integers $\mathbb{Z}\subseteq \mathcal{O}_K \subseteq \mathcal{O}_L$. If $0 \neq \mathfrak{p} \trianglelefteq ...
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6answers
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The Langlands program for beginners

Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things he must know before he can understand the Langlands program and its ...
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1answer
54 views

Proving an equivalence relation, $a^3\equiv\pmod 9$, in $\mathbb{Z}[\omega]$

I would appreciate help with two steps in solving this problem (self-study) from Ireland & Rosen (3.25) The problem states: Let $\lambda= 1-\omega \in \mathbb{Z}[\omega]$. And $a\equiv 1\pmod ...
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1answer
407 views

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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0answers
241 views

Does every ring of integers sit inside a ring of integers that has a power basis?

Given a finite extension of the rationals, K, we know that $K=\mathbb{Q}[\alpha]$ by the primitive element theorem, so every $x \in K$ has the form $$x = a_0 + a_1 \alpha + ... + a_n \alpha^n$$ with ...
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3answers
57 views

Show $a+bi \equiv 0,1 \pmod{1+i}$

This is a problem (self-studier) from Ireland & Rosen (3.23): Show $a+bi \equiv 0,1 \pmod{1+i}$ with $a,b \in\mathbb{Z}$. There is an extensive hint/solution offered, some parts of which I would ...
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0answers
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Proof of Chapter 2 Proposition 2.6a in Silverman Arithmetic of Elliptic Curves

The following is Proposition 2.6(a) in Silverman's AEC: "Let $\phi: C_1 \rightarrow C_2$ be a nonconstant rational map of smooth (projective) curves over an algebraically closed field $K$. Then ...
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1answer
45 views

confusion with calculating the ideal class group of a quadratic field

I am a bit confused with the procedure of calculating the ideal class group of a quadratic field. From what I understood the computation starts by finding the Minkowski's bound say $n$. Then we list ...
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1answer
78 views

When is $\mathbb{Z} [x]/f(x) $ a Dedekind domain?

Given a monic separable irreducible polynomial $f$ with integer coefficients, when $\mathbb{Z} [x]/f(x)$ is a Dedekind domain? And when it happens to be a Dedekind domain, how to know its class ...
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1answer
40 views

Ring of algebraic integers

Let $R$ be a ring of algebraic integers, $p ∈\mathbb{Z}$ a prime integer. Then the set $\mathcal A$ of all prime ideals $P ⊂ R$ such that $P ∩ \mathbb{Z}= p\mathbb{Z}$ is finite and nonempty. Also, ...
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1answer
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Find all points of finite order on the elliptic curve $y^2+7xy=x^3+16x$.

I am studying Rational Points on Elliptic Curves by Silverman and Tate. This is Problem 2.12 (h). Determine all of the points of finite order on the elliptic curve $y^2+7xy=x^3+16x$. Also ...
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0answers
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Rational prime with specified factorization in $\mathbf{Z}[\mu_q]$

Let $r$ and $f$ be given positive integers. Prove that there exist primes $p$ and $q$ such that $p\mathbf{Z}[\mu_q]$ (where $\mu_q$ is a primitive $q$th root of unity) is a product of exactly ...
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2answers
64 views

The irreducibility of $ X^q - 3 $ in the field extension $ \mathbb{Q}(\zeta_q, 2^{1/q}) $

Old question: I am not sure whether this statement is even true, although it "feels" as if it should be. The statement is as stated in the title: $ X^q - 3 $ is irreducible over the splitting field $ ...
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0answers
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Evaluating the norm of an ideal by considering integral basis for $\mathcal{O}_K$

There is this trick in my lecture notes that I don't think is correct. Let $K=\mathbb{Q}(\sqrt{-5})$ then my notes say that the ideal $(2,1+\sqrt{-5})$ in $\mathcal{O}_K$ has obviously norm $2$ since ...
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1answer
26 views

Results in algebraic number theory regarding ramified split and inert primes in quadratic fields

I am currently reading some notes in algebraic number theory but they are not really self contained and I am guessing the following results must hold. Let $K$ be a quadratic field and consider the ...
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1answer
483 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 $R$ ...
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0answers
40 views

Prime ideals of the ring of integers lying over $p\mathbb{Z}$ [closed]

Let $A$ be the ring of all elements of $\mathbb{C}$ that are integral over $\mathbb{Z}$, and $p\in\mathbb{Z}$ a prime element. Are there infinitely many prime ideals of $A$ lying over $p\mathbb{Z}$? ...
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0answers
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Intuition for Adeles and Ideles

I'm currently studying some class field theory and read about the notion of adeles and ideles. However, the object seems a bit arbitrary to me; is there a natural way to think about the adele-ring? ...
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possible norms of prime ideals in the class group of $K=\mathbb{Q}(\sqrt{-21})$

I have an example in my notes where we try to compute the class group of the quadratic field $K=\mathbb{Q}(\sqrt{-21})$. My notes then proceed to evaluate the Minkowsk's bound< ...
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A Problem from Marcus' Number Fields

I have been stuck on the 17th problem of the 3rd chapter from Marcus' Number Fields. Let $K=\mathbb{Q}[\sqrt-23]$ , $L=\mathbb{Q}[\omega]$ where $\omega = e^{2.\pi.i/23} $ . Let $P$ be one of the ...
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1answer
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Why assuming that the ideal in Minkowski's bound is prime

Minkowski’s bound states that given a quadratic field $K(\sqrt{d})$ then every class of ideals in $\mathcal{O}_K$ contains an integral ideal of norm<$\lambda(d)$. Then my notes say that this ...
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3answers
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How can one express $\sqrt{2+\sqrt{2}}$ without using the square root of a square root?

I was trying to review some analysis, and came across problem 3 from page 78 of Walter Rudin's Principles of Mathematical Analysis. As part of the problem, I wanted to try to write ...
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1answer
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Definition of split prime in quadratic fields

I have the following definition of split prime number $p \in \mathbb{Z}$ in my lecture notes that I don't understand. Let $K$ be a quadratic field, the definition I have says: $p$ is called split in ...
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0answers
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Prime Decomposition of an ideal in a number field.

I have been stuck on the 26th problem of the 3rd chapter from Marcus' Number Fields. Let $\alpha=\sqrt[3]{m}$ where $m$ is a cubefree integer, $K=\mathbb{Q}[\alpha]$, $R=\mathbb{A} \cap ...
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0answers
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Maximal $\mathbb Z_p$-orders in $\mathbb Q_p[G]$

Let $G$ be a finite group and $S$ a finite set of prime numbers. I know that every separable $\mathbb Q$-algebra $A$ contains a maximal $\mathbb Z$-order but I wonder if the following is true. ...
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3answers
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Are these quotient modules isomorphic?

Let $K$ be an algebraic number field and $\mathcal{O}_K$ its ring of integers. For a non-zero ideal $\mathfrak{a}$ of $\mathcal{O}_K$ and an element $c \in \mathcal{O}_K \setminus \{0\}$ I wonder ...
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0answers
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The tower of ramification indices

Let $K\subset L\subset M$ be an extension of number fields. Let $R\subset S\subset T$, be their algebraic integers rings, respectively. Suppose that $P\subset R$, $Q\subset S$, $U\subset T$ be a prime ...