Questions related to the algebraic structure of algebraic integers

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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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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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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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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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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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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
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formal derivative algebraic [on hold]

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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1answer
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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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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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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
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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
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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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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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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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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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1answer
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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
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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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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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1answer
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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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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
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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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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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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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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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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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2answers
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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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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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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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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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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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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 ...
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Geometric Proof that Gaussian Integers form PID

In my reading on Algebraic Number Theory, I came across a description of a proof that the Gaussian Integers form a PID- you select a nonzero element $\alpha$ of minimal norm in the ideal, where the ...
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Stronger form of Hensel's lemma?

Let $f \in \mathbb{Z}_p[x]$ and suppose $|f(a)|_p < |f'(a)|_p^2$ for some $a \in \mathbb{Z}_p$. Let $a_1 = a$, and for $n \ge 1$ let$$a_{n+1} = a_n - f(a_n)/f'(a_n).$$How do I see that this defines ...
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Definition of $S$-ideles

This is a basic notational question. Let $K$ be a number field and $M_K$ the set of all places of $K$ with $S\subset M_K$ a finite subset. Write $\mathfrak J_K$ for the idele group of $K$ and ...
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Arithmetic in a dihedral extension

Let $\;$L = $\Bbb Q$[$\sqrt[4]{2}$, i ]$\;$ which is a dihedral extension of the rationals. There are three quadratic and five quartic intermediate fields between L and $\Bbb Q$. The following ...
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1answer
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Basis of a Cyclotomic Field

I've started learning algebraic number theory when I found something that confused me; for a prime $p$, where $\zeta=e^{(2\pi i/p)}$, a primitive $p$-th root of unity. Then the extension ...
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1answer
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Exists sequence converging to $0$ in $\mathbb{R}$, $1$ in $\mathbb{Q}_2$?

Does there exist a sequence of elements $x_1, x_2, x_3, \ldots$ of elements of $\mathbb{Q}$ that converges to $0$ in $\mathbb{R}$ and converges to $1$ in $\mathbb{Q}_2$?
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Constructing Witt polynomials

I am reading about Witt vectors, and I keep seeing the following set of congruences often: For example, in these notes here, on page 3 we see the following congruences: \begin{align} ...
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Galois group is isomorphic to group of characters. What can we say then?

Let $\overline{K}/K$ denote the separable closure of a finite field $K$ of characteristic $p$ and let $\mu_{n}$ denote the group of $n$-th roots of unity in $\overline{K}$, where $(n,p)=1$. Let us ...
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An ideal of $\mathcal O_K$ [duplicate]

Let $K$ be a number field and let $\mathcal O_K$ be its ring of integers. Let $I$ be a non-zero ideal in $\mathcal O_K$. If $I$ is free as a $\mathcal O_K$-module then is it a principal ideal? ...
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If $K\cap\Bbb Q^{\text{cycl}}=\Bbb Q(\zeta_m)$ and $K/\Bbb Q$ Galois, then $\text{Gal}(K(\zeta_n)/K)\cong\text{Gal}(\Bbb Q(\zeta_n)/\Bbb Q(\zeta_m))$

$\DeclareMathOperator{\Gal}{Gal}$ Here is my argument: Induction on the number of primes dividing $n/m$. If there are two primes (i.e., $K(\zeta_n) = K(\zeta_{q_1},\zeta_{q_2})$, where ...
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Double cosets (Neukirch's Algebraic Number Theory)

This is a question from Neukirch's Algebraic Number Theory, Ch.1 $\S$9. Let $A$ be a Dedekind domain with quotient field $K$, $L$ a finite separable field extension of $K$ and $B$ the integral ...
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2answers
58 views

Show that $\mathbb{Q}(\sqrt{-14},\sqrt{-2\sqrt{2}-1})$ has degree 8 over $\mathbb{Q}$

We know that the extension $\mathbb{Q}(\sqrt{2\sqrt{2}-1}$) over $\mathbb{Q}$ has degree 4 by considering the minimal polynomial mod 3. Now I want to show that $-14$ isn't a square in this field. How ...
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Showing reducibility of a polynomial in a Discrete Valuation Ring

Let $R$ be a complete discrete valuation ring with uniformiser $\pi$. I would like to show that a polynomial $f$ in $R[X]$ is reducible. Does it suffice to show that $f$ is reducible in ...
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2answers
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ramification index in an example

Let $L=\mathbb{Q}_5[x]/(x^4+5x^2+5)$, where $\mathbb{Q}_5$ is the field of 5-adic numbers. Note that the polynomial that we are quotienting out by is an Eisenstein polynomial. So $L/\mathbb{Q}_5$ is a ...
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1answer
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Neukirch Abstract Kummer Theory. Understanding a Proof.

This question is a sort of follow up to this question, where I introduced context. Neukirch mysterious homomorphism in Abstract Kummer Theory (in his book ANT) The thing I don't understand know is ...
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$p$-adic field with infinite residue field. [duplicate]

I am reading J M Fontaine's book where on page 7 the following definition is made: A local field($K$) is a complete discrete valuation field whose reside field($k$) is a perfect field of ...
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1answer
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Algebraic number theory, Marcus, Chapter 3, Question 9

Question 9 in Marcus book. Let $K$ and $L$ be the number field such that $K\subset L$ and let $R,S$ be their algebraic integers, respectively. a) Let $I$ and $J$ be ideals in $R$, and suppose ...
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Norm of element $\alpha$ equal to absolute norm of principal ideal $(\alpha)$

Let $K$ be a number field, $A$ its ring of integers, $N_{K / \mathbf{Q}}$ the usual field norm, and $N$ the absolute norm of the ideals in $A$. In some textbooks on algebraic number theory I have ...