Questions related to the algebraic structure of algebraic integers

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0
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0answers
19 views

Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers upto $\frac{1}{2}p^2_x -2$ exvert all non-trivial $\Xi_x$-classes?

This question follows from the information provided below. Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers up to $\frac{1}{2}p^2_x -2$ exvert all non-trivial ...
13
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2answers
110 views
+300

Decomposition of an algebraic number into a sum or product of algebraic numbers with smaller degree

An algebraic number can be identified by its minimal polynomial together with isolating intervals with rational bounds for its real and imaginary parts. The degree of an algebraic number is the degree ...
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0answers
42 views

Norm restricted to $\mathbb Q$

Consider an algebraic extension $K$ of $\mathbb Q$ and suppose that on $K$ we have a non trivial norm $||\cdot||$. Is it possible to have that the restriction $||\cdot||_{\mathbb Q}$ is the trivial ...
0
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1answer
24 views

Incipit of chapter VI of Neukirch's ANT book.

The title of the chapter VI of the neukirch's ANT is "Global class field theory", and the first few lines are the following: the author doesn't explain what is $K$ here, but from the previous ...
3
votes
2answers
80 views

Kummer Theory - Example of Subgroup of $K^{*}$ containing $K^{*m}$ for global fields.

I am trying to understand Kummer theory and I wish to apply it to global fields, so our field $K$ containing $\mu_m$ should be $\mathbb{Q}(\zeta_m)$. Let $B$ be a subgroup of $K^{*}$ containing ...
10
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0answers
177 views
+200

What does the ideal norm of matrix elements really mean?

Say we have a number field $K$ (specifically, an imaginary quadratic field) and a $2\times2$ matrix $\sigma=\pmatrix{a&c\\b&d}$ with elements $a,b,c,d\in\mathcal O_k$, the ring of integers of ...
2
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0answers
12 views

Comparing Dedekind zeta functions

It is known that non-isomorphic number fields can share the same Dedekind zeta function. However, there don't appear to be any examples of very low degree so in these cases the zeta function must ...
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0answers
12 views

“linear independence” of characters? Artin's theorem.

Let $G$ be a monoid and $K$ be a field. We define a character of $G$ in $K$ as a monoid homomorphism $f\colon G\to K^{\times}$, where $K^{\times}$ denotes the multiplicative group of $K$. One fact ...
8
votes
2answers
162 views

What is going on in this degree 8 number field that fails to be a quaternion extension of $\mathbb{Q}$?

This is a soft but very mathematically hands-on question. Hopefully it will be interesting to more than just me. Thanks in advance for your help in thinking clearly about what follows. I have been ...
4
votes
1answer
60 views

Norm on “tower” of fields (the question comes from algebraic number theory)

Consider the field extensions $L\supset K'\supset K$, where both $L|K$ and $K'|K$ are finite and Galois. I want to prove that $$N_{L|K}(L^\ast)\subseteq N_{L|K'}(L^{\ast})$$ Maybe it is very easy, ...
3
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1answer
49 views

Help with Proposition 13.2.9 in Ireland and Rosen

I'm currently self studying Ireland and Rosen's A Classical Introduction to Modern Number Theory and got stuck on the proof of Proposition 13.2.9. In this proof, $p$ is a prime not dividing $m$, $D, ...
2
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1answer
41 views

Subrings of the product of rings of algebraic integers

Let $K,L$ be a number fields and $\mathcal{O}_K,\mathcal{O}_L$ their rings of algebraic integers, and $n_K=[K:\mathbb Q]$, $n_L=[L:\mathbb Q]$. Suppose that $R$ is a subring of ...
2
votes
0answers
40 views

Fixed fields in Neukirch's book (chap. IV): notational problem

I am reading chapter IV of Neukirch's ANT, and there is a thing that I don't understand. First of all I have to introduce the notations of chapter IV. $G$ is a profinite group and: Clearly this ...
17
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7answers
3k views

Beginner's text for Algebraic Number Theory

What's good book for learning Algebraic Number Theory with minimum prerequisites? Assume that the reader has done an basic abstract algebra course.
8
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3answers
1k views

Book(s) Request to Prepare for Algebraic Number Theory

I would appreciate suggestions for books to enhance my learning in algebra so as to be able to read Samuel's "Algebraic Theory of Numbers" and eventually at least begin Neukirch's "Algebraic Number ...
4
votes
3answers
2k views

Pre-requisites needed for algebraic number theory

I acknowledge my limited knowledge of abstract algebra(My background comprising groups and subgroups from Herstein's Topics in Algebra is hardly worth mentioning). And yet, I confess I really like ...
3
votes
2answers
37 views

Quadratic integer ring with universal side divisor?

It seems that in every paper mentioning universal side divisors, they are defined very succinctly and with a bunch of symbols, so that I remain completely confused as to what they are and how to find ...
2
votes
1answer
54 views

Equivalent condition for being a regular prime ideal

$\newcommand{\p}{\mathfrak{p}}$ $\newcommand{\tp}{\tilde{\mathfrak{p}}}$ $\newcommand{\tA}{\tilde{A}}$ I have a question about Neukirch, Algebraic Number Theory, page 92. The problem is to show the ...
2
votes
1answer
443 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 $R$ ...
13
votes
1answer
198 views

CFT via Brauer groups vs via ideles

I am interested in the relationship between the following two versions of CFT: Version 1: (Brauer Group Version) Let $K$ be a number field. One constructs, for every finite place $v$ of $K$, a map ...
2
votes
1answer
51 views

Is $\mathbb{Z}[\sqrt{-7}]$ a UFD or not? [duplicate]

Is $\mathbb{Z}[\sqrt{-7}]$ a UFD or not? I feel like there should be some easy example showing that it is not, but I can not think of one.
4
votes
2answers
57 views

Is $\phi :SL_2(Z) \to SL_2(Z/NZ)$ still surjective if we replace Z with some ring of integers?

It is well known that the natural map $\phi :SL_2(Z) \to SL_2(Z/NZ)$ is surjective. So that the kernel, i.e. the principal congruence subgroup is of finite index. But what if we replace Z with some ...
6
votes
1answer
66 views

Does the determinant give you the index over $\mathcal{O}_k$ as well as over $\mathbb{Z}$?

It is a standard fact that if $M$ is a nonsingular $n\times n$ integer matrix, the index of the $\mathbb{Z}$-span of its columns as an abelian group in $\mathbb{Z}^n$ is $|\det M|$. What happens if we ...
2
votes
2answers
91 views

Completion of the unit group of a local field

Let $K$ be a number field and $\mathfrak{p}$ a finite prime of $K$. Denote the unit group of the ring of integers of the local field $K_\mathfrak{p}$ (i.e. the completion of $K$ via $\mathfrak{p}$) by ...
5
votes
2answers
124 views

In what quadratic or quartic integer ring is a prime of the form $a^4 + 4^b$ guaranteed to split?

The obvious choice seemed at first to be $\mathbb{Z}[\root 4 \of 4]$. But since I know next to nothing about quartic fields, I thought to look in the quadratics. For the first few such primes in ...
0
votes
1answer
26 views

An isomorphism between product of number fields, contains the same number of factors [on hold]

Suppose that we have an isomorphism of rings $$f:K_1\oplus\cdots\oplus K_r\to K'_1\oplus\cdots\oplus K'_s,$$ with $K_i$'s and $K_j'$'s are a number fields, the sum and the product are componentwise. ...
1
vote
3answers
50 views

Local reciprocity map applied to norm

This question concerns the local reciprocity homomorphism $r_L : L^\times \to G_L^\text{ab}$, where $L$ is a local field with absolute Galois group $G_L$. If $K\subset L$ is a subfield and $r_K$ is ...
2
votes
1answer
26 views

A question concerning archimedean places on number fields

Let $L$ be a number field and $G=Aut(L\mid \mathbb{Q})$. Let $|\cdot|$ be the usual archimedean value on $\mathbb{C}$ and, by abuse of notation, its restriction to $\mathbb{Q}$. Then the archimedean ...
1
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1answer
29 views

Property of multiplication of ideals in $\mathcal{O}_K$

Let $\mathfrak{a}, \mathfrak{b}$ be two coprime ideals of $\mathcal{O}_K = \mathbb{Z}[\sqrt{-d}]$ such that $\mathfrak{a}\mathfrak{b} = (n)$ for some $n \in \mathbb{Z}$. Does $\mathfrak{a}^m = (u)$ ...
3
votes
1answer
40 views

Absolute Galois group $\text{Gal}(\overline{K}/K)$ of any number field $K$ has a non-open subgroup of any prime index $p$?

Let $K$ be a number field, and let $p$ be a prime number. Does $G = \text{Gal}(\overline{K}/K)$ necessarily have a subgroup of index $p$ that is not open?
3
votes
1answer
35 views

Easy computational example of first cohomology group: Is this how we do it?

I'm an undergraduate student learning a little bit of group cohomology on my own. I'd like to compute a few examples of the low-dimensional cohomology groups in some special cases to get some ...
7
votes
1answer
88 views

Upper bound on exact power of wild prime that divides the different

Let $K$ be a number field and let $p\in \mathbb{Z}$ be a prime. Suppose that $p\mathcal{O}_K=Q^eI$, with $\gcd(Q, I)=1$ and let $\mathcal{D}_{K/\mathbb{Q}}$ be the different ideal of $K$. We know that ...
2
votes
2answers
33 views

Primitive elements of number fields which span rings of integers

My question is the following: given a number field $K$, does there exist a primitive element $\alpha$ of $K$ over $\mathbb{Q}$ such that the ring $\mathcal{O}_K$ of integers of $K$ is isomorphic to ...
1
vote
2answers
114 views

Artin reciprocity theorem for Hilbert class field

In Cox's book "Primes of the form $x^2 + ny^2 $..." gives the following statement of Artin reciprocity theorem, for the Hilbert class field (i.e. maximal unramified Abelian extension) Artin's ...
3
votes
1answer
32 views

Outer Automorphisms of Galois groups

Assume $K/\mathbb{Q}$ is Galois extension of number fields and let $F \subset K$. Let $Gal(K/F) \cong G$ and let $\sigma$ be an outer automorphism of $G$ as an abstract group. Is there a way to ...
2
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1answer
489 views

How to prove a generalized Gauss sum formula

I read the wikipedia article on quadratic Gauss sum. link First let me write a definition of a generalized Gauss sum. Let $G(a, c)= \sum_{n=0}^{c-1}\exp (\frac{an^2}{c})$, where $a$ and $c$ are ...
2
votes
1answer
28 views

What are the equivalent statements of GRH using the Möbius or Liouville functions?

We all know that Riemann Hypothesis can be stated as properties of $\mu$ or $\lambda$, particularly in terms of the random behaviour of those functions with "square root" bounds. Are there similar ...
16
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3answers
507 views

Galois Group of the Hilbert Class Field

Let $K/\mathbb{Q}$ be a number field with Galois group G and let $L/K$ be the Hilbert class field of $K.$ It is easy to show that $L$ is Galois over $\mathbb{Q}$ and I am interested in knowing this ...
4
votes
1answer
73 views

Inverse Galois theory and Hilbert class field

I am not sure if the following questions have an answer. (Question 1) Let $G$ be a finite Abelian group. Is it possible to find an unramified Abelian extension $L/K$ such that $$G \cong ...
6
votes
1answer
120 views

$\mathbb{Q}(\sqrt[3]{17})$ has class number $1$

Let $\alpha:=\mathbb{Q}(\sqrt[3]{17})$ and $K:=\mathbb{Q}(\alpha)$. We know that $$\mathcal{O}_K=\left\{\frac{a+b\alpha+c\alpha^2}{3}:a\equiv c\equiv -b\pmod{3}\right\}.$$ I have to show that $K$ has ...
2
votes
2answers
120 views

Hilbert class field of cubic field

Let $K=\mathbb Q(\sqrt[3]7) $ be a pure cubic field with class number 3. I want know how to compute its Hilbert Class Field. I know that its degree of extension is 3. Thank You in advance.
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1answer
47 views

Algebraic integers divided by a prime

Denote the set of all algebraic integers by $A$ (thus $A$ is a subset of $\mathbb C$). Denote by $A'$ the set of all $\alpha\in A$ such that $\frac{\alpha}{p}\not\in A$ for all primes $p$. Question : ...
1
vote
1answer
62 views

Norm in the cyclotomic integers

Let $\alpha \in \mathbb{Z}[\zeta_3]$, where $\zeta_3=e^{2\pi i/3}$ is a cube root of unity. So $\alpha=x+y\zeta_3$ for $x,y\in\mathbb{Z}$. Show that the norm $N(\alpha)$ can be written as ...
2
votes
2answers
50 views

Prime $\mathfrak{p} \in$ Max$(\mathbb{Z}[\sqrt{10}])$ splits completely iff principal

L.S., This is an exercise from my lecture notes on algebraic number theory: Let $L = \mathbb{Q}(\sqrt{2},\sqrt{5})$ and $K = \mathbb{Q}[\sqrt{10}]$. Prove that prime ideal $\mathfrak{p} \in$ ...
0
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0answers
26 views

Are there separable polynomials in $K[Y][X]$ with constant discriminant?

Let $A$ be a ring and $P\in A[X]$ be a monic degree $n$ polynomial. Let $Disc(P):= Res(P,P')$ be the discriminant of $P$. If $A=\mathbf Z$, then Minkowski's theorem says that there are no non ...
2
votes
1answer
63 views

Goldbach Conjecture, what are new research methods after Chen's work?

For Goldbach Conjecture, my understanding is that there are three major methods to attempt it: Schnirelmann density circle method sieve method (Chen used two parameter sieve method to get his ...
5
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1answer
77 views

A specific problem on Class field theory

Let $K$ be a quadratic complex number field. Let $p$ be a prime greater than $5$ unramified in $K/\mathbb{Q}$. Let $M$ be the compositum of all finite $p$- extensions of $K$ which are unramified ...
2
votes
1answer
77 views

Abelian extensions with squarefree discriminant

Is it true that for all $2 \leq n \in \mathbb N$ we can find an abelian extension of the rationals with squarefree discriminant and degree n? Are there even infinitely many? Some even degrees may be ...
2
votes
2answers
36 views

Completion of an extension $K|\mathbb{Q}$ w.r.t. a non archimedean abs. value, isomorphic to $K\cdot \mathbb{Q}_p$

As the title suggests I want to prove the following result: given $\mathfrak{p}|(p)$, prime ideal of $\mathcal{O}_K$ over $p$, we have the canonical absolute value induced by it on $K$, and we can ...
1
vote
1answer
20 views

Bounds for zeta function residue

Let $K$ be an algebraic number field and let $c = c(K)$ denote the residue at $s = 1$ of its zeta function. It is known Wikipedia: class number formula that c can be determined via $$c = \frac{2^r ...