Questions tagged [algebraic-number-theory]

Questions related to the algebraic structure of algebraic integers

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Let $D,m$ be relatively prime integers with $m$ odd. Then $D \equiv 0,1 \pmod 4$ and $D \equiv b^2 \pmod m$ implies that $D \equiv b^2 \pmod{4m}$

This is from a proof in David A. Cox's Primes of the Form $x^2+ny^2$: Lemma 2.5 Let $D\equiv 0,1 \bmod 4$ be an integer and $m$ be an odd integer relatively prime to $D$. Then $m$ is properly ...
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What does the notation "$(R, +, \cdot)$" and "$(C, +, \cdot)$" mean? (From a book on Number Systems.)

The number "$i$" also is as much of a mental construct and no more , as the number "$s$". The new number is defined in such a way that it not only satisfies $i^2=-1$, but when ...
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ramified in prime degree cyclic extension implies totally ramified in prime power cyclic extension

Let $K/\mathbb Q$ be a prime power cyclic extension, say of degree $p^n$. If a prime $q$ ramifies in the subfield $K_1$ of $K$ such that $[K_1:\mathbb Q]=p$, can one assert that $q$ is totally ...
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What am I misunderstanding in the definition of this subgroup?

Let $K$ be a number field with $d \in \mathcal{O}_K \setminus \{0\}$, $\sqrt{d} \notin \mathcal{O}_K$. Consider the group of units $\mathcal{O}_K[\sqrt{d}]^\times$ and denote conjugation of elements ...
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What is the value group of $ \Bbb{F}_q((t^{1/p^n}))$ ?(Detailed caluculation)

What is the value group of $ \Bbb{F}_q((t^{1/p^n}))$ ?(Detailed caluculation) I'm not good at calculating value group, and my only tactics is to calculate the value of each element. $|t^{p^{1/n}}u|=1/...
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2 votes
1 answer
37 views

Ramification in cyclotomic fields

Let $m$ be a positive integer and $K_m$ be the $m$-th cyclotomic field. It is well known that a prime $p$ is ramified in $K_m$ if and only it divides $m$ and in the particular case when $m=p^r$, it is ...
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Galois cohomology and Galois representation- Related areas

Are there any interesting results we can get by combining the field of Galois cohomology and the field of Galois representations? Well, these two fields are sort of mathematical languages deals with ...
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How to simplify the expression for Dirichlet inverse of $\varphi$ further?

$\varphi$ : Euler totient function $\mu$: M$\ddot{o}$bius function $I =\chi_{\{1\}}$ $N\in\Bbb{K}^{\Bbb{N}}$ : $N(n)=n$ $u $ : unit function i.e $u(\Bbb{N}) =\{1\}$ We know $\varphi(n) =\sum_{d|n}\mu(...
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Is integer part of $e^n$ infinitely often even and odd?

Let $a_n:=\lfloor e^n\rfloor$. Are $\{n\mid 2|a_n\},\,\{n\mid 2|a_n-1\}$ both infinite sets? More generally, for any irrational number $\alpha>1$, is each set of the form $\{n\mid p|\lfloor\alpha^n\...
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What is the perfect closure of $ \Bbb{F}_p((t))$ and its value group?

What is the perfect closure of $ \Bbb{F}_p((t))$ and its value group ? I think perfect closure is $ \Bbb{F}_p((t^{1/p^∞}))$, and I think if so, $\bigcup_{n\geqq1}(1/p^n) \Bbb{Z}$? $ \Bbb{F}_p((t^{1/p^...
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Tate gamma factor as a principal value integral

Let $F$ be a local field, $\chi$ a multiplicative character of $F^{\ast}$, and $\psi$ an additive character of $F$. The gamma factor $\gamma(s,\chi,\psi)$ is defined by means of the local functional ...
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2 votes
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Proof regarding principal factors of the discriminant in $\mathbb{Q}(\sqrt{d})$

So I understand there are (up to $\pm$) exactly two primitive (no rational integer factors) elements $\alpha_1 ,\alpha_2 \in \mathcal{O}_K$ such that the fundamental unit $\varepsilon$ of $K=\mathbb{Q}...
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1 answer
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For every prime p there is a sum of squares congruent to -1 mod p [duplicate]

For every prime $p$, there exists $a,b \in \mathbb{Z}$ such that $p\mid a^2+b^2+1$ For context, this question shows up as a statement on a hint to showing that every positive integer is a sum of 4 ...
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In search for methods for finding a closed form for the roots of a non-homogenous diophantine equation.

I'm trying to solve a special case of a number theoretical problem, and it relies on finding a closed form for the roots of a non-homogenous diophantine equation with four variables, but I could only ...
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2 votes
1 answer
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Bhargava’s proof of van der Waerden conjecture: how to use Hilbert irreducibility to show almost all polynomials have Galois group $S_n$

On the first page of his paper where he proves van der Waerden’s conjecture, Bhargava mentions that Hilbert’s irreducibility theorem shows that the number of monic integer polynomials of degree $n$, ...
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Valuation ring of completion of a field

I am actually quite confused. I have done an exercise that $\mathbb{Z}_p$ is a completion of $\mathbb{Z}$ w.r.t. the $p$-adic norm. Then again I got to know after reading somewhere that $\mathbb{Z}_p$ ...
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A non-cyclic quartic

Given the quartic $x^4 - 2c x^3 + (c^2 - d^2) x^2 + 2a^2 c x - a^2 c^2 = 0$ for integers $a$,$b$ and $c$ where $d^2 = a^2 + b^2$. Looking at the Galois group $G$ of this quartic, we can WLOG assume ...
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Equivalent Conditions for a Prime to not Contain Conductor

I am working through some notes on algebraic number theory, and am trying to show the following five conditions are equivalent. Here, $A\subset B$ is an extension of Dedekind domains corresponding to ...
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2 votes
1 answer
98 views

Why ${\rm Gal}\left (\overline{\mathbb{Q}_p}/ \mathbb{Q}_p\right )$ action on $ \overline{\mathbb{Q}_p}$ extends to its action on $\mathbb{C}_p$?

Why ${\rm Gal}\left (\overline{\mathbb{Q}_p}/\mathbb{Q}_p\right )$ action on $ \overline{\mathbb{Q}_p}$ extends to ${\rm Gal}\left (\overline{\mathbb{Q}_p}/ \mathbb{Q}_p\right )$ action on $\mathbb{C}...
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van der Waerden's proof that a monic $p(x) \in \mathbb{Z}[x]$ has Galois group $S_n$ with probability 1

This paper mentions that van der Waerden proved some results on the density of monic integer polynomials with Galois group the symmetric group $S_n$ in 1936. I have found van der Waerden's original ...
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2 votes
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Can one show that in a certain sense, "most" polynomials have Galois group $S_n$? [duplicate]

Intuitively, it seems that given a random irreducible $p(x) \in \mathbb{Q}[x]$ of degree $n$, the Galois group of $p(x)$ over $\mathbb{Q}$ should be $S_n$. Otherwise, if $\alpha_1,...,\alpha_n$ are ...
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  • 3,506
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why $|a|=|\sigma (a)|$, where $|\cdot|$ denotes absolute value on $\overline{ \Bbb Q_p}$?

Let $a\in\overline{ \Bbb Q_p}$, $\sigma\in Gal$( $\overline{ \Bbb Q_p}/ \Bbb{Q}_p$), then, why $|a|=|\sigma (a)|$, where $|\cdot|$ denotes absolute value on $\overline{ \Bbb Q_p}$? I think we should ...
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  • 398
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57 views
+100

What is the local basis at $0$ of inverse limit topology?

What is the local basis of inverse limit topology at $0$? For example, $\mathbb Z_p=\lim\mathbb Z/p^n\mathbb Z$ has $$\{ \{(\cdots,α_0)\in\mathbb Z_p|α_m=・・・=α_o=0\} \mid m≧0\}$$as a local basis at $0$...
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Totally ramified $\mathbb{Z}_p$ extension

Let $K$ be a $p$-adic field, that is, $K$ is of characteristic $0$ and the residue field is perfect of characteristic $p$. On many places, it writes let $K_\infty$ be a totally ramified $\mathbb{Z}_p$-...
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1 vote
1 answer
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Units in ring of integers of $5^{th}$ cyclotomic field

Let $K=\mathbb{Q}(\omega)$ with $\omega$ a primitive $5^{\text{th}}$ root of unity. I'm trying to prove that $$ \mathcal{O}_K^\times=\left\{\pm\omega^{a}\left(\frac{1}2+\frac{\sqrt{5}}2\right)^b: a,b\...
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1 vote
0 answers
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Calculating the fundamental units of integer ring using submodules

Suppose we have some extension $K/\mathbb{Q}$, where $K = \mathbb{Q}(\theta)$ and some submodule $\mathbb{Z}[\theta'] \subseteq \mathcal{O}_K$. Is there a general method to calculate $[\mathcal{O}_K^\...
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5 votes
0 answers
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Compute the Fourier expansion of adelic Eisenstein series associated to the classical holomorphic Eisenstein series.

For each place $v$ of $\mathbf{Q}$, define $\Phi_v:(\mathbf{Q}_v)^2\to\mathbf{C}$ by $$ \Phi_v(x,y)=\begin{cases} \mathbb{I}_{\mathbf{Z}_v}(x)\mathbb{I}_{\mathbf{Z}_v}(y)&\text{if $v<\infty$},\\...
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2 votes
0 answers
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Is there a way to evaluate de natural density of numbers of the form $ax^2-by^2$?

I was trying to evaluate the density of numbers of the form $5x²-y²$ and ended up pondering about this question. Can we always find the natural density of numbers of the form $ax^2-by^2$, where $(a,b)=...
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3 votes
1 answer
144 views

Showing that the only integer solutions to $y^3=4x^2+47$ are $x=250,-250$

We’ve computed that the class number for $\mathbf Q(\sqrt{-47})$ is $5$. From my attempt at this question, we have the equation of ideals $$(y)^3=(2x+\sqrt{-47})(2x-\sqrt{-47}).$$ Next we show that ...
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1 answer
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Prime ideals of subrings of the ring of Gaussian integers

Can anyone give me a hint with proving the following, Let $\alpha\in\mathbb Z[i],$ and let $P$ be a non-zero prime ideal of $\mathbb Z[\alpha].$ Show that the quotient $\mathbb Z[\alpha]/P$ is a ...
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1 vote
1 answer
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Infinitely large Galois extensions of $\mathbb Q$ inside $\mathbb Q_p$

Let $p$ be a fixed prime number and denote $\mathbb Q_p$ the field of $p$-adic numbers. For each positive integer $n$, I would like to construct a finite Galois extension $K/\mathbb Q$ of degree at ...
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0 votes
1 answer
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Sets without Dirichlet/polar density

Let $K$ be a number field with norm $N=N_{K/\mathbb{Q}}$, $\mathcal{P}$ its set of prime ideals, and $A\subset\mathcal{P}$. We say that the Dirichlet density is $$\delta^D(A):=\lim\limits_{s\to1^+}\...
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1 vote
0 answers
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$p^{th}$ power residue symbol for $p$-free numbers that are coprime to $p$.

Recall that if a number field $K$ contains the primitive $p^{th}$ root of unity $\zeta_{p}$, then for every prime ideal $\mathfrak{p}$ of $K$ coprime to $p$ and every $\mathfrak{p}$-adic unit $\alpha$ ...
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1 vote
0 answers
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Neukirch's description of $\mathcal{O}(X)$ in Localization chapter

In page 66 of Algebraic Number Theory Book by Neukirch,he defines $A(X)$ as the localization from integral domain $A$ with multiplicative subset $S$ as complement of $\bigcup_{P\in X} P $ where $P$ ...
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1 answer
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Definition of fractional ideal

I have a little problem with the definition of a fractional ideal. The definition I've been given is a set $f\subseteq Q=\text{Frac}(R)$ such that $\exists b\in R\backslash \{0\}$ such that $b.f\...
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-1 votes
0 answers
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How to prove $\mathcal{O}_{\mathbb{C}_p}/ p \mathcal{O}_{\mathbb{C}_p}=(\mathcal{O}_{\mathbb{C}_p}/ p \mathcal{O}_{\mathbb{C}_p})^p$

Let $\mathcal{O}_{\mathbb{C}_p}$ the ring of integers of complex $p$-adic numbers $\mathbb{C}_p$ which are defined as completion of alg closure $\overline{\mathbb{Q}_p}$ with respect $\vert \cdot \...
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1 answer
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value group of $E=\Bbb{Q}_p(p^{1/e})$

I want to find what is a value group of $E=\Bbb{Q}_p(p^{1/e})$($e$ is positive integer, and this is totally ramified extension of degree $p$). I know value group of $K= \Bbb{Q}_p$ is {$p^a$|$a∈\Bbb{Z}$...
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0 votes
1 answer
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$(O_K/pO_K)^p=O_K/pO_K$holds, then $∃b∈K^×$, such that $|a-b^p|≦|p|$

Let $L$ be finite extension of $ \Bbb{Q}_p$ and field $K$ satisfies $L⊆K⊆\Bbb{C}_p$.Let $O_K$ be ring of integers of $K$. Suppose $(O_K/pO_K)^p=O_K/pO_K$・・・① holds, then $∃b∈K^×$, such that $|a-b^p|≦|...
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0 votes
1 answer
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Soft Question: Introductory books on Algebraic Number Theory

I am a high school student in Britain. I have recently studied elementary number theory and absolutely loved it. I got as far as to prove results like quadratic reciprocity, Fermat's sum of two ...
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2 votes
1 answer
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Isomorphism between $K\otimes_{\mathbb{Q}}\mathbb{R}$ and $K_{\mathbb{R}}$, Minkowski theory

This is a basic question from Neukirch’s book ‘Algebraic Number theory’, chapter 1, $\S$5. Let $K$ be a number field, $K_{\mathbb{C}}$ the $\mathbb{C}$-vector space $\prod_{\tau}{\mathbb{C}}$ where $\...
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6 votes
1 answer
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A convergence lemma for adelic zeta function in automorphic forms

I'm reading Godement-Jacquet's classic Zeta functions of simple algebras (1972, Springer). On page 153, the first line: " We also $\textbf{take for granted}$ the following lemma (numbered 11.3)......
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1 answer
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The discriminant and Stickelberger's Theorem

Let $\theta$ be a root of the polynomial $x^3-x+2$, which is irreducible. Consider the basis $\{1,\theta,\theta^2\}$, which is clearly a rational basis for the integer ring in $Q(\theta)$. Now, I'm ...
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Intersection of $ \Bbb{R}$ and $ \Bbb{Q}_p$ [duplicate]

Intersection of $ \Bbb{R}$ and $ \Bbb{Q}_p$ For distinct prime $p$ and $q$,intersection of $ \Bbb{Q}_p$ and $ \Bbb{Q}_q$ is just $ \Bbb{Q}$, but what about $ \Bbb{Q}_p$ and $ \Bbb{R}$ ? $ \Bbb{R}$ is ...
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  • 584
0 votes
0 answers
67 views

real quadratic fields with 2 ramified primes

I have observed that when considering real quadratic fields with class number 1 and 2 having 2 ramified primes, the first ramified prime was (almost always, like 98% of the time) congruent to $3 \bmod ...
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0 votes
1 answer
65 views

Why $p$-adic logarithm is continuous on $\mathbb{Q}_p^\times$?

Neukirch defined, in Algebraic Number Theory, Neukirch (5.4) p. 136, the $p$-adic logarithm on $1 + p\mathbb{Z}_p$ as $\log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots$ Then, he extends this ...
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1 vote
2 answers
61 views

Does $ \Bbb{Q}_2$ has $\sqrt{-1}$?

Does $ \Bbb{Q}_2$ has $\sqrt{-1}$? I tried to use Hensel lemma as usual. Let $f(x)=x^2+1$. But if some $a∈\Bbb{Z}$, $f(a)=0$, then $f'(a)$ can always divide by $2$. So I cannot use Hensel lemma. Could ...
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2 votes
1 answer
72 views

Number theory True or false [closed]

I have two questions that I need to determine if they are true or false and I'm unsure (a) There exists a number field $K$ such that $\mathcal{O}_{K} \cong \mathbb{Z[x]}$ (b) If $K = \mathbb{Q(\sqrt{3}...
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0 votes
1 answer
136 views

Sufficient condition of given extension is not unramified

Let $p$ be a prime number. If we want to judge whether $p$ is prime element of $ \Bbb{Q}_p(α)$ for some fixed element $α∈\overline{ \Bbb{Q}_p}$, what is the basic strategy? To find a ring of integers ...
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  • 398
0 votes
1 answer
41 views

Show that the sum of 2 algebraic integers is an algebraic integer using resultants

I've started working on algebraic numbers very recently for a memoir, that is I didn't study them in class. I need them, and particularly algebraic integers, to prove a couple propositions which aren'...
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2 votes
0 answers
88 views

Is $1+p/2!+p^2/3!+p^3/4!・・・$ convergent in $ \Bbb{Z}_p$?

Let $p$ be an odd prime.Is $1+p/2!+p^2/3!+p^3/4!・・・$ convergent in $ \Bbb{Z}_p$? I know $1+p+p^2/2!+p^3/3!+・・・$ converges but what about the titled case ?
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