Questions related to the algebraic structure of algebraic integers

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“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 ...
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1answer
53 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, ...
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1answer
39 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 ...
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0answers
29 views

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
38 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 ...
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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, ...
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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 ...
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2answers
55 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 ...
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1answer
65 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 ...
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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. ...
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1answer
25 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 ...
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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?
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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 $\mathfrak{a}^m = (u)$ ...
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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 ...
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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 ...
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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 ...
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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 : ...
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1answer
50 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.
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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 ...
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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
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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 ...
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0answers
25 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 ...
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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 ...
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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 ...
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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 $\mathfrak{p} \in$ ...
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0answers
33 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 ...
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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 $K$. ...
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0answers
18 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 ...
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0answers
18 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 ...
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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 ...
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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 ...
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2answers
49 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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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3answers
65 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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2answers
39 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 $V_k\to V_k\times V_K,\ x\mapsto(x,x^{-1})$. ...
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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 ...
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2answers
31 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 ...
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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 ...
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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?
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1answer
51 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
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1answer
31 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 ...