Questions related to the algebraic structure of algebraic integers

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1answer
58 views

Integral and prime ideal in Dedekind domain

Let $A$ be an Dedekind domain, $K$ its quotient field, $L$ a finite separable extension of $K$, and $B$ the integral closure of $A$ in $L$. If $p$ is a prime ideal of $A$, then $pB$ has a ...
8
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4answers
112 views

What are some applications of Chebotarev Density Theorem?

Let $L/K$ be a Galois extension of number fields and let $\mathcal{C}$ be a conjugacy class in $Gal(L/K)$. Let $\mathbb{P}(K)$ be the set of all prime ideals in $K$ and let ...
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0answers
43 views

Finding a finite Galois cover trivializing a lisse sheaf

Given a lisse $\mathbb{F}_{\ell^r}$-sheaf on a smooth curve $U$ defined ofer $\bar{\mathbb{F}_p}$, Katz says here, in p. 33, that $\mathcal{F}$ "becomes constant on some connected finite etale galois ...
2
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0answers
43 views

Decomposition into product of prime ideals

Let $K=\Bbb{Q}(a)$, where $a$ satisfies $a^3 - 5a + 5 = 0$. For $n\le7$, I need to compute explicitly the decomposition of $n\Bbb{Z}_K$ as a product of prime ideals. Here $\Bbb{Z}_k$ stands for the ...
3
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0answers
150 views

Irreducibility of a polynomial modulo infinitely many primes

Suppose $f(x) \in \mathbb{Z}[x]$ is an irreducible monic polynomial over $\mathbb{Q}(\alpha)$ of degree $n$, where $\alpha$ is a root of a monic polynomial $g(x) \in \mathbb{Z}[x]$. Assume that the ...
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0answers
16 views

Infinite Galois theory: every subgroup of finite index is open. Proof-check

Let $\Omega/k$ be a (possibly infinite) Galois extension and let the group $G=G(\Omega/k)$ be equipped with the Krull Topology. My question is about a statement I think its true, but I'm not sure. ...
2
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1answer
36 views

Is there -0 value?

I'm a very newbie (if not ignorant) to most math-related topics, but ever since I started my primary school, I was always told, that there is no such thing like "minus zero" value, because zero is the ...
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0answers
41 views

Number of square-free polynomials over a finite field - a combinatorial interpretation?

One can show using zeta functions that the number of (monic)square-free polynomials of degree $n$ over a finite field $\Bbb F_q$ is $q^n-q^{n-1}$. For instance, this is done here. The answer is ...
3
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1answer
53 views

Show that the prime ideals above a prime $p$ are principal

Let $K=\mathbb Q(\alpha)$ where $ \alpha^3 -5\alpha + 5 = 0 $. I need to show that the prime ideals above 5 are principal, and find a generator for them. I have worked out the prime decomposition of ...
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0answers
33 views

How to tell if a set is open in the Krull topology?

I'm an undergraduate not very familiar with topology trying to understand the so called Krull Topology in the context of infinite Galois Theory. We proceed as follows: Let $\Omega/k$ be a (possibly ...
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2answers
91 views

Irreducibles elements in $\mathbf Z[\sqrt{-3}]$

The ring $A:=\mathbb Z[\sqrt{-3}]$ is the prototype of the rings usually used in a first algebraic number theory course to show the difference between prime and irreducible elements. I was wondering ...
7
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1answer
68 views

“Lifting” fibres of morphism of arithmetic schemes to get rid of “nongeometric” ramification

This is a soft question and really a request for pointers towards a certain rigorous formulation of geometric intuition I've had for some "arithmetic schemes". I'm looking for ideas and key references ...
2
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2answers
53 views

Ramification of primes in cyclotomic field

I don't know a lot of algebraic number theory, but I think the following is true: let $E/\mathbf{Q}$ be an algebraic number field, and $p\in\mathbf{Z}$ a prime. Then there a unique integers $e,f,g$ ...
3
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0answers
64 views

Self Learning — Number Theory

I was wondering if there were any good online courses/lecture videos (preferably courses/videos but books would work too) for self learning algebraic number theory. I have seen sites like MIT ...
2
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1answer
56 views

Compute the decomposition of $5\mathbb{Z}_K$ as a product of prime ideals

Let $K = \mathbb{Q}(\alpha)$ such that $\alpha^3 - 5\alpha + 5 = 0$. It is easy to show that $\mathbb{Z}_K = \mathbb{Z}[\alpha]$ and that $5$ is not maximal in $\mathbb{Z}[\alpha]$. So we cannot use ...
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1answer
30 views

Showing $A[\theta] \subseteq B \subseteq \frac{1}{d} A[\theta]$, where $A$ is a Dedekind domain

Suppose we have $A$ a Dedekind domain, $K$ its field of fractions. Let $L$ be a field extension of $K$ of degree $n$ and also that it is separable. Let $B$ be the integral closure of $A$ in $L$. ...
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2answers
34 views

Showing $B_P$ is a finitely generated module over $A_P$ where $P$ is a prime ideal in a Dedekind domain.

Suppose we have $A$ a Dedekind domain, $K$ its field of fractions. Let $L$ be a field extension of $K$ of degree $n$ and also that it is separable. Let $B$ be the integral closure of $A$ in $L$. ...
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2answers
31 views

Is the set of real algebraic numbers in $(0,1)$ the same as the set of fractional parts of real algebraic numbers in $(1, \infty )$?

It seems that way to me, but I'm not sure how to prove it rigorously. Say, we have the number $x>1$ that is a root of some polynomial with integer coefficients: $$a_0+a_1 x+a_2 x^2+\dots +a_n ...
6
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1answer
75 views

Prove that if $n \in \mathbb{Z}[\sqrt{2}]$ has an even norm, then $\sqrt{2} \mid n$

Aside from multiplying and dividing some specific numbers in this ring, e.g., $(1 + \sqrt{2})\sqrt{2}$ I have not really done anything productive on this question. I either go around in circles or ...
3
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1answer
58 views

Higher Ramification Groups for $\mathbb{Q}(\sqrt{d})|\mathbb{Q}$. Clever way to compute

I'm asked to computing the higher ramification group for quadratic extensions $K=\mathbb{Q}(\sqrt{d})|\mathbb{Q}$. They are defined as follows, for a prime ideal $\mathfrak{p}$, ...
3
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0answers
70 views

Algebraic numbers and geometric series - from finite to infinite, similarity with transcendental numbers

It started with a game I played with the inverse of golden ratio, but now I have some questions about the connection of infinite geometric series and algebraic numbers. The example first. Since we ...
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0answers
17 views

Dimesion of an affine variety- solution verification

I have affine variety $V=\{(t,t^2,t^3)|t\in\mathbb{Q}\}$ and if I'm right I have $I(V)=(y-x^2,z-x^3)$ and coordinate ring is $\mathbb{Q}[V]\cong \mathbb{Q[t]}$ (I set $x=t$). I have this definiton : ...
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1answer
34 views

Ramification of primes under cetain conditions

Let $K\subset L=K(\gamma)$, ($\gamma$ an algebraic integer) be number fields such that there exists a $k\in \mathcal O_K$ and some $n\in\Bbb N$, $\gamma^n=k$. Also, there exists an ideal $I\subset ...
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0answers
128 views

Swapping the digits of an algebraic number (e.g. $\sqrt 2$)

Let an algebraic number, say $ a=\sqrt 2 = 1.41421356237309504880...$, and define $$b=f(a)=1.14243165323790058408...$$ by swapping the digits $a_{2i+1}$ and $a_{2i+2}$ for $i≥0$, corresponding to ...
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1answer
37 views

Euler totient function and unramified extension of $\mathbb{Q}_p$. A clarification.

I'm studying the construction of unramified extensions, and many references say that it's enough to attach the $p^n-1$ primitive root of unity to $\mathbb{Q}_p$ in order to obtain the unique degree ...
0
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0answers
18 views

System of rational polynomial equations with complex root also has a solution of algebraic numbers [duplicate]

Consider a system of equations $$f_1(x_1,...,x_k)=0,...,f_n(x_1,...,x_k)=0$$ where $f_1,...,f_n$ are polynomials in $\mathbb{Q}[x_1,...,x_k]$. Suppose the system has a solution in $\mathbb{C}^k$. ...
1
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1answer
54 views

What are the invariants of a number field? [closed]

How is defined an invariant of a field? Given a certain field extension $L/K$, is it related with the Galois group ${\rm Gal}(L/K)$? In the case of number fields, which are the invariants associated ...
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0answers
42 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 ...
2
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1answer
62 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
32 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 ...
2
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0answers
27 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 ...
0
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0answers
32 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 ...
4
votes
1answer
74 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, ...
2
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1answer
53 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 ...
15
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2answers
211 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 ...
3
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1answer
57 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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votes
2answers
61 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 ...
4
votes
2answers
79 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 ...
7
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1answer
86 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
33 views

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

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. ...
2
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1answer
29 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
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1answer
59 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
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1answer
32 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
46 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
96 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
41 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
49 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
84 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
134 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 ...
3
votes
1answer
45 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 ...