Questions related to the algebraic structure of algebraic integers

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4
votes
2answers
49 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 ...
4
votes
1answer
57 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 ...
1
vote
1answer
21 views

An isomorphism between product of number fields, contains the same number of factors

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. ...
6
votes
1answer
46 views

Non-Euclidean domains examples which have universal side divisor

Let $R$ be a ring. A nonzero nonunit element of $R$ is called a universal side divisor if for every element $x$ of $R$ there is some element $z$ of R such that $u$ divides $x - z$ in $R$ where $z$ is ...
2
votes
1answer
24 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 ...
3
votes
1answer
36 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?
1
vote
1answer
28 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 $\mathfrak{a}^m = (u)$ ...
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 ...
7
votes
1answer
83 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 ...
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 ...
1
vote
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 : ...
2
votes
1answer
49 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.
5
votes
2answers
121 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 ...
2
votes
1answer
32 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 ...
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 ...
0
votes
0answers
24 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 ...
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 ...
2
votes
2answers
34 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 ...
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 $\mathfrak{p} \in$ ...
1
vote
0answers
31 views

The sum of all primitive $n$-th roots of unity in the algebraic closure of $F(x)$

If $n$ is any natural number, then $\mu(n)$ is the sum of all primitive $n$-th roots of unity in $\mathbb{C}$ where $\mu$ is the Möbius function defined as $\mu(n)=0$ if $n$ is not square free and ...
1
vote
1answer
25 views

Completion of a Galois Extension $K | \mathbb{Q}$ w.r.t. to a non archimedean abs. value is a Galois extension

As the title suggests, I'm interested in a proof of the following fact: Let $K | \mathbb{Q}$ be a finite Galois extension, and let $\mathfrak{p}$ be a prime ideal of the ring of integers of $K$. ...
1
vote
0answers
16 views

Generalisation of Fermat's Little Theorem in Function Fields

There is a well-known generalisation of Fermat's Little Theorem as for any $a\in\mathbb{Z}$ and $n\in\mathbb{N}$ we have $$\sum_{d\mid n} \mu(n/d) a^d\equiv 0\pmod n.$$ where $\mu$ is the Möbius ...
0
votes
0answers
17 views

What is Dirichlet class number formula for d when d is NOT a fundamental discriminant?

According to Wikipedia, Dirichlet published a proof of the class number formula for quadratic fields in 1839, but it was stated in the language of quadratic forms. Let d be a fundamental ...
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 ...
3
votes
3answers
80 views

Why the terminology “global fields” and “local fields”

Let $p$ be a prime number. A global field is defined as a finite extension of $\mathbb Q$ or $\mathbb F_p(t)$. On the other hand one can show that a local field (which is by definition a complete ...
1
vote
2answers
47 views

On the relative discriminant of a cyclic extension of an algebraic number field whose relative degree is a prime number

Let $K$ be a cyclic extension of an algebraic number field $k$ whose relative degree is a prime number $l$. Hasse wrote(see below) in his "Bericht" that the relative discriminant of $K/k$ is of the ...
1
vote
1answer
30 views

Is this ring a Dedekind domain?

Let $p(x)\in \mathbb{Z}[x]$ be a monic, irreducible polynomial in $\mathbb{Z}[x]$. For the field $K:=\mathbb{Q}[x]/(p(x))$, its ring of algebraic integers $\mathcal{O}_K$ is always a Dedekind domain ...
1
vote
1answer
35 views

Can class number $h(d)$ equal to zero for some $d$?

We know that $L(1, \chi)$ is related to the class number $h(d)$ with a constant. And this is one way that we can prove $L(1, \chi)$ not vanish on $s = 1$. What confused me is: we know that class ...
2
votes
0answers
29 views

Quantitative aspect of Chebotarev Density Theorem

I recently learned the Chebotarev Density Theorem for global fields. As far as I have seen, all applications of CDT seem to focused around proving some set of prime ideals (or places in function ...
0
votes
1answer
41 views

Is there an explicit formula for the expression $(a\mathbb{Z}+b) \cap (a'\mathbb{Z}+b'),$ not involving $\cap$?

Thinking of $\mathbb{Z}$ as a ring, the ideals of $\mathbb{Z}$ are precisely those subsets of the form $a\mathbb{Z}.$ Hence intersections of ideals can be computed by taking lowest common multiples. ...
6
votes
0answers
92 views

Discrete valuation fields and representation as power series

Let $(K,v)$ be a discrete valuation field ($v$ is surjective). Let $\mathcal O$ be the ring of integers of $v$ and moreover let $\mathfrak p$ be the unique maximal ideal of $\mathcal O$. Then we have ...
1
vote
1answer
24 views

eigenvalues of sum of matrices with algebraic integers eigenvalues

Let $A, B$ be two matrices such that they both have all eigenvalues in $\mathbb{A}$, the ring of algebraic integers. The question is: it is true that the matrix $A+B$ does have all of its eigenvalues ...
4
votes
3answers
63 views

Unique factorization theorem in algebraic number theory

Consider the set $S = a + b \sqrt {-6}$, where $a$ and $b$ are integers. Now, to prove that unique factorization theorem does not hold in set $S$, we can take the example as follows: $$ 10 = 2 \cdot ...
3
votes
1answer
52 views

p-adics $\mathbb{Q}_p$ is a field if and only if $p$ is a power of a prime

I want to show that the ring $\mathbb{Q}_p$ is a field for any prime $p$, so I want to show that every nonzero element has an inverse. I thought of the following argument, but I can't seem to locate ...
2
votes
3answers
46 views

Show that the number $z=\sqrt[3]{4}-2i$ is algebraic, that is satisfied a polynomial equation with integer coefficients.

Show that the number $z=\sqrt[3]{4}-2i$ is algebraic, that is satisfied a polynomial equation with integer coefficients. I thought I could use the Fundamental theorem of Algebra, but it seems to be ...
1
vote
0answers
35 views

What does $\log^3$ stand for in this paper by K. Győry?

In this paper from 1980 K. Győry proved an upper bound for the absolute value of the solutions to a given Thue-Mahler equation, but I don't understand what he meant with $\log^3$. Namely, consider a ...
1
vote
1answer
18 views

What are the dual polyhedra of the face-centered cubic lattice?

For a given lattice $L$ we could define the set of points closest to one point more than any other. $$ \{ x \} = \min_{\ell \in L} \|x - \ell \| \in \mathbb{R}^3$$ This generalizes the "fractional ...
3
votes
2answers
37 views

Importance of the compactness of idele group

Si $k$ is a number field and $J_k$ is the group of $V_k^*$ of invertible elements of the adele ring $V_k$ with the induced topology given by the morphism $V_k\to V_k\times V_K,\ x\mapsto(x,x^{-1})$. ...
1
vote
1answer
54 views

On the genus of a curve and its set of rational points.

The genus $g$ of a nonsingular curve $C$ of degree $n$ is defined as $g = \frac{1}{2}(n-1)(n-2)$. Let $C(Q)$ denote the set of rational points on $C$. By Faltings, we know that $C (Q) < \infty$ for ...
1
vote
2answers
28 views

A polynomial with integer coefficients with prescribed set of roots modulo infinitely many primes

Given an infinite set of distinct prime numbers $p_1, p_2,\ldots,p_n,\ldots$, and arbitrary integres $a_1, a_2,\ldots,a_n,\ldots$, is there a nonzero polynomial $f(x)$ with integer coefficients such ...
1
vote
3answers
46 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 ...
0
votes
0answers
24 views

Monic polynomial terminology

If the constant term of a monic polynomial is one or negative one, is there a name for that special kind of monic polynomial?
3
votes
1answer
45 views

Ramification of primes in number fields in terms of valuations

Let $L/K$ be a number field extension. Is there a definition of ramification of primes(both infinite and finite) in terms of the valuations induced? This answer gives a definition for infinite ...
2
votes
1answer
30 views

Factorization of ideal in field $\mathbb{Q}(\sqrt[3]{2})$ and its normal closure

So far I've worked only with quadratic fields, and I'm not sure how to work with 3rd roots. I have ideal $(5)$ and need to factor it in $\mathbb{Q}(\sqrt[3]{2})$ and its normal closure. I know that ...
6
votes
1answer
118 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 ...
7
votes
1answer
45 views

Product of “Fake”-Galois Conjugates

My apologies if this question ends up being a duplicate; I did my best to search for an answer, but I have no idea what to call this stuff I'm working with, so I couldn't really find much. There is a ...
0
votes
1answer
29 views

What is the intersection of $\mathbb Z$ with the ideal generated by $1-\zeta_n$?

For example, 1-(-1) is in the ideal <2>, whenever $n$ is even. Suppose $R=\Bbb Z[\zeta_n]$, and (by the above), that $n$ is odd. We know 1+$\zeta_n$ can be multiplied by ...
0
votes
1answer
46 views

Rational points on $4x^5 + y^2 = z^2$

Does the title curve have any nonzero rational points ? I have to admit that i didn't find any significant insight to this problem.
0
votes
0answers
28 views

What are the pre-requisites required to understand Milnor's book on algebraic K- theory?

I want to understand Steinitz’ theorem on the structure of finitely generated modules over Dedekind domains. I also want to have some general awareness regarding what Algebraic K-theory is about. ...
4
votes
1answer
76 views

Square and cubic roots in $\mathbb Q(\sqrt n)$

Here is my question : Let $n$ a squarefree positive integer, $m \ge 2$ an integer and $a+b \sqrt n \in\mathbb Q (\sqrt n).$ What (sufficient or necessary) conditions should $a$ and $b$ satisfy so ...