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Questions tagged [algebraic-number-theory]

Questions related to the algebraic structure of algebraic integers

2
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0answers
20 views

Ring of integers of $\mathbb{Q}(i,\sqrt{5})$

I'm trying to find the ring of integers $A_L$ of $\mathbb{Q}(i,\sqrt{5})$. I know that the ring of integers of $\mathbb{Q}(i)$ is $\mathbb{Z}[i]$ and that the one of $\mathbb{Q}(\sqrt{5})$ is $\mathbb{...
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0answers
12 views

Open problems related to modular objects

What are some open problems in the theory of modular forms, mock modular forms, L-functions, quantum modular forms and all our little modular friends ! If you have interesting research papers to share ...
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1answer
26 views

Proof explanation of $P(p^e) \cong \mathbb{Z}/(p-1)\mathbb{Z} \times \mathbb{Z}/(p^{e-1})\mathbb{Z}$.

The following is from Classical Theory of Algebraic Numbers by Paulo Ribenboim : $P(p^e)$ is the set of all nonzero residue classes a modulo m, where gcd(a, m) = 1. My question underlined and in ...
2
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0answers
40 views

(Milne CFT p. 36) Showing an $A$-module in an exact sequence is cyclic.

I am currently working on the following proof from Milne's Class Field Theory. Here $K$ is a nonarchimedean local field, $A$ its valuation ring $A = \{\alpha \in K: | \alpha| \leq 1\}$. So $A$ is a ...
5
votes
1answer
64 views

Lubin-Tate formal groups are $p$-divisible groups

I am trying to understand how to see whether a given formal group is $p$-divisible. Let $A$ be a complete noetherian local ring with maximal ideal $\mathfrak{m}$ and residue field $k$ of ...
4
votes
2answers
34 views

$p=a^2+ab+41b^2$ iff $-163$ is a quadratic residue

Prove that a prime $p$ can be written as $p=a^2+ab+41b^2$ iff $-163$ is a quadratic residue modulo $p$. What I have in mind is something like this: look at $\mathbb{Q}[\sqrt{-163}]$ which has the ...
1
vote
1answer
24 views

Quotients of ramification groups seen as additive subgroups of $S/Q$.

I am working through Marcus's book $Number$ $Fields$, and I have been working for a while on exercise $22$ from chapter $4$. Letting $K \subset L$ be a Galios extions of a number field $K$, and let $...
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0answers
19 views

Dedekind theorem of ramification in cyclotomic fields

Let $\Bbb Q(w)$ denote the $n$-th cyclotomic field then$\Bbb Z[w]$ is its ring of integers, $d$ denotes the discriminant of $\Bbb Q(w)$, and $p\mid d$ then $p\Bbb Z$ must ramify in $\Bbb Q(w)$. In ...
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0answers
15 views

Sequential continuity of automorphisms of number fields

If I have some Galois number field $K$, a sequence of elements $(x_n)_{n=1}^{\infty}\subset K$ converging to some $z\in K$, and an automorphism $\sigma$ of K, when does $\lim\limits_{n\to\infty}(\...
2
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1answer
71 views

Find a number field whose unit group is isomorphic to $\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}$

Find a number field whose unit group is isomorphic to $\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}.$ I'm trying to use Dirichlet's Unit Theorem to solve this problem. It states that if $K$ is a number ...
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1answer
19 views

Show that any non-zero prime ideal of $R$ is invertible.

Theorem $:$ Let $R$ be an integral domain such that $R$ is Noetherian, integrally closed and every non-zero prime ideal of $R$ is maximal with quotient field $K.$ Then every non-zero prime ...
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1answer
31 views

Help finding error in equating $L$-series

Let $$s(n) = \#\{(x,y) \in \mathbb{Z}^2 : x^2 + 3y^2\ = n\}$$ for $n \geq 1$ an integer and let $K = \mathbb{Q}(\sqrt{-3})$. The class number of $K$ is $1$ and the ring of integers $\mathcal{O}_K$ ...
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2answers
54 views

Books of algebraic number theory [duplicate]

I am learning algebraic number theory, the exercises are so hard for me, could you please recommend me a book with answers? Many thanks!
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0answers
20 views

Irreducibility and reducibility [closed]

Is there is any universal method or law which can be applied to check reducibility and irreducibility of a given polynomial in $\mathbb{Q}[X]$?
2
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1answer
46 views

Marcus Number fields exercise 17 chapter 4

I know this question has already been posted, but I don't manage to understand the comments. Exercise $17$ $(e)$ on Marcus' Number Fields, Chapter $4$ My problem is exactly the same as the one ...
1
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1answer
20 views

Proving surjectivity to define inertia group

I'm having trouble understanding the proof of propsoition 2.6.14 of these notes, in particular the part about surjectivity. Since $\bar{\sigma}(\theta)$ is also a root of $h$, it is a root of $...
3
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0answers
31 views

Langlands L functions for groups over finite fields.

In some reading on automorphic/Langlands-related papers I have seen some authors refer to the finite field analogues of Langlands objects, such as admissible representations, L factors but a simple ...
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1answer
35 views

A polynomial whose range gaps are the primes [closed]

Not sure what to add here. I responded to hardmath in TN e Peter. If you feel it doesn't generate discussion I will rethink the question and read the purpose of your platform and try to reform the ...
1
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1answer
34 views

extension of discrete valued fields and ring of integers

Let $K$ be a complete discrete valued field with ring of integers $A$, maximal ideal $m$ and uniformizer $\pi_K$. Let $L$ be a finite extension of $K$ with ring of integers $B$, maximal ideal $m'$ ...
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0answers
30 views

About sums similar to gauss sums

It is well known the case for sums like: $$ \sum_{i=0}^{p^n -1}\zeta^{-ai}, $$ where zeta is a primitive $p^n$-rooth of $1$. But, is there a standard formula for sums like: $$ \sum_{i=0}^{p^n -1}i^...
2
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1answer
31 views

Neukirch's interpretation of unit group and class group

In page 22 from Neukirch's Algebraic Number Theory, he defines the class group $Cl_K$ of a number field $K$ to be the quotient of group of fractional ideals $J_K$ by the subgroup of principal ideals $...
3
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2answers
33 views

Understanding a solution for the fact that $\mathcal{O}_{\mathbb{Q}(\sqrt[3]{2})}=\mathbb{Z}[\sqrt[3]{2}]$

The problem above has many answers in StackExchange. I'm trying to understand this specific one. He mentions the formula $\text{disc}(\mathbb{Z}[\alpha])=(\mathcal{O}:\mathbb{Z}[\alpha])^2\text{disc}(...
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0answers
24 views

Inertia fields and decomposition fields in cyclotomic extensions

I've been trying to figure out the following problem. Given a prime $p$, and $\zeta_m = e^{2\pi i /m}$, and the factorization $m = p^kn$ with $(p,n) = 1$, then we know that the Galois group of $\...
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0answers
27 views

Two number fields with isomorphic Galois groups but different Galois closure of their maximal real subfields

$\newcommand\Q{\mathbb Q}$Suppose $G$ is a transitive finite permutation subgroup of $S_{2n}$ having fixpoint-free involution in and outside its center. Suppose furthermore that I know that there is a ...
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1answer
33 views

On the Newton polygon for Laurent series

I'm stuck with an understanding of what should be the Newton polygon for a Laurent series. I'm reading ''An introduction to G-function" by Dwork and he dedicates only three pages to Newton polygons ...
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0answers
54 views

Can we find the exact sum of series $\sum_{n=0}^\infty \frac{1}{(n!)^n}$ [closed]

Can we find the exact sum of series $\sum_{n=0}^\infty \frac{1}{(n!)^n}$? We know thaf the sum is $e$ without that power 'n' in the denominator.
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2answers
171 views

How did Gauss conjecture there were nine Heegner numbers?

Coming from someone not very knowledgable in algebraic number theory it seems odd. At the time they didn't have the computing power to determine whether very high values (>>163) were Heegner numbers; ...
5
votes
0answers
58 views

Can two monic irreducible polynomials over $\mathbb{Z}$, of coprime degrees, have the same splitting field?

Let $f,g \in \mathbb{Z}[X]$ be monic polynomials. It is possible for distinct monic polynomials over $\mathbb{Z}$ to have the same splitting field. For example $f = x^4 - 2$ and $g= x^4+2$ both have ...
0
votes
1answer
24 views

Bounds on representations via (non-positive) binary quadratic forms

Suppose that for some $n\in \mathbb{Z}$ we know that there are $x,y\in \mathbb{Z}$ such that $x^2-dy^2=n$ for some $d\in \mathbb{N}$. Can we say anything about how large $x$ and $y$ are compared to $n$...
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0answers
24 views

$L/K$ is finite separable field extension with $O_v$ valuation ring of $K$ w.r.t valuation $v$. Integral closure of $O_v$ in $L$ is DVR?

Let $L/K$ be a finite separable field extension with $O_v$ valuation ring of $K$. Here I will not assume $O_v$ complete and similarly for $K$ as well.(i.e. There is a completion $\hat{O}_v$ of $O_v$ ...
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0answers
45 views

$p$-adic analytic function bounded implies coefficients bounded?

Let $K$ be a complete valued subfield of $\mathbb{C}_p$. Let $\mathcal O=\{z \in \mathbb C_p \colon \vert z \vert \leq 1\}$ be the ring of integers in $\mathbb C_p$ and $\mathfrak m=\{z \in \mathbb ...
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1answer
65 views

When is $2\cos(\pi \frac{i}{n}) + 2\cos(\pi \frac{j}{m}) = -1 $

Let $a,b$ be algebraic integers, i.e., the roots of polynomials with coefficients in $\mathbb{Z}$ and the leading coefficient is $1$. Assume now that $a + b$ is equal to a rational integer--more ...
1
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1answer
67 views

Prove that the quotient ring is not a PID

Show that $R = \mathbb{C}[x,y]/(y^2-x^3-1)$ is not a PID. My idea is to find an ideal of $\mathbb{C}[x,y]$ containing $(y^2-x^3-1)$ and show that its image is not principal. So I have $J = (x,y+1)$, ...
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1answer
35 views

Showing linear independence of power basis of $\mathbb{Q}(\sqrt[4]{3})$ over $\mathbb{Q}$.

I have been told that $\mathbb{Q}(\sqrt[4]{3})$ as a vector space over $\mathbb{Q}$ has dimension $4$. If $\alpha = \sqrt[4]{3}$, then I am guessing a basis is $1, \alpha, \alpha^2, \alpha^3$. I can ...
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0answers
16 views

Why I can't make question on stack exchange app? [migrated]

Whenever I try to upload a question .It is showing that your limit is exhausted .Try after 7 days. . Can anyone suggest me how to come up this problem. And ask my doubts frequently .
1
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1answer
60 views

Is this element in the ring of integers?

Let $\alpha = \sqrt[3]{2}$, which is integral, and $K = \mathbb{Q}(\alpha)$ be a number field of degree $3$, with basis ${\{1, \alpha, \alpha^2}\}$ and let $\mathcal{O}_K$ denote its ring of ...
3
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0answers
36 views

What's the relation between class group and class field theory?

It's clear for me what is the relationship and difference between a group and a field in abstract algebra. But, I just started with algebraic number theory and it is not clear to be how (ideal) class ...
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1answer
72 views

Squares in $\mathbb{Z}_p$

Consider the integral binary quadratic form $$f(x,y) = 2Axy+Bx^2$$ with $A,B \in \mathbb{Z}$ different from $0$. In Cassel's book "Rational quadratic forms" page 237 he claims that for $p \neq 2$ ...
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votes
1answer
14 views

Does any fractional ideal of $R$ always contain a non-zero element of $R$?

Let $R$ be an integral domain. Let $A$ be a non-zero fractional ideal of $R.$ Then can we say that $A$ always contains a non-zero element of $R$? Please help me in this regard. Thank you very much.
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0answers
26 views

Understanding the Hurwitz-Kronecker Class Number Formula

The Hurwitz-Kronecker Class Number is given by the formula $H(d)=\sum_{Q\in Q_d/(\Gamma=PSL_2(\mathbb{Z}))}\frac{1}{w_Q}$ where $w_Q=card(stab(\alpha_Q))$ with $\alpha_Q$ being the unique zero ...
3
votes
1answer
28 views

Showing that ideals are principal in ring of integers of cubic field

Let $K = \mathbb{Q}(\sqrt[3]{6})$. Factorise $\langle p \rangle$ into prime ideals in $\mathcal{O}_K$ for $p = 2, 5, 13$, checking that the factors are principal. I used the Dedekind-Kummer Theorem ...
3
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0answers
38 views

Abelian extensions and the Artin map

I'm trying to understand a proof of the following: Let $L$ and $M$ be Abelian extensions of $K$ (a number field). Then $L\subset M$ if and only if there is a modulus $\mathfrak{m}$, divisible ...
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2answers
39 views

Trace of an algebraic integer is an integer?

Let $F$ be a number field and let $\alpha \in F$. If $\alpha \in \mathcal{O}_F$, then it is known that $N(\alpha) \in \mathbb{Z}$. I was wondering if something similar can be said about the trace? ...
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0answers
15 views

history of analogy between number field and function fields

it is well known that an analogy exist between number fields and function fields and you can translate ideas and problems about one of them to another. there are many problems in number theory ...
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0answers
39 views

Viewing ramification in $p$-adic fields in the context of algebraic geometry

Let $f: X \rightarrow S$ be a morphism of schemes which is locally of finite type. We say $f$ is unramified if for every $x \in X$, we have $(\Omega_{X/S})_x = 0$, where $\Omega_{X/S}$ is the sheaf ...
2
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0answers
43 views

Splitting primes and quadratic reciprocity.

I'm reading through Marcus's wonderful book Number Fields, and I had two questions on his proof of quadratic reciprocity in chapter $4$. Marcus states first that if $p$ is an odd prime, and $q$ is ...
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2answers
70 views

How come algebraic closure is not unique?

I was getting confused trying to understand why an algebraic closure of a field is not unique. If I consider any rational polynomial in one variable, then the roots may not be in $\mathbb{Q}$ but aren'...
2
votes
1answer
41 views

Showing integrality of a certain ring [duplicate]

Let $A\subset B$ be integral domains, and let $b\in B$ be a unit. I am trying to show that $A[b]\cap A[b^{-1}]$ is integral over A. Let $x\in A[b]\cap A[b^{-1}]$. It suffices to show that $A[x]$ is a ...
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votes
2answers
118 views

Confused about the splitting of 2 in $\mathbb{Q}(i).$

How do we split 2 in the cyclotomic ring $S = \mathbb{Z}[\sqrt{-1}]?$ Clearly the field $\mathbb{Q}(i)$ is a degree 2 normal extension. However, $(2)$ in $S$ is equal to $(2) = (1 - i)^2.$ Thus, the ...
3
votes
0answers
108 views

Next book in learning Algebraic Number Theory

I have just finished the book Introductory Algebraic Number Theory by Kenneth S. Williams and Saban Alaca. My aim is to reach to graduate level to do research, especially in one or more of the ...