Questions related to the algebraic structure of algebraic integers

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3
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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
votes
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 ...
4
votes
0answers
32 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 ...
6
votes
2answers
86 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
votes
1answer
67 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
votes
2answers
52 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
votes
0answers
63 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
votes
1answer
55 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 ...
1
vote
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 $L$. ...
1
vote
2answers
33 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 $L$. ...
1
vote
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
votes
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
votes
1answer
55 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
votes
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 ...
0
votes
0answers
16 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 : ...
1
vote
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 ...
8
votes
0answers
123 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 ...
2
votes
1answer
36 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
votes
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
vote
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 ...
0
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0answers
41 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
votes
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
votes
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
votes
0answers
26 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
votes
0answers
30 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
72 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
votes
1answer
50 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
votes
2answers
208 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
votes
1answer
54 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, ...
5
votes
2answers
59 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
78 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
votes
1answer
83 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 ...
0
votes
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
votes
1answer
28 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
58 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
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
42 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
38 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
48 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
78 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
132 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
44 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
75 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 ...
1
vote
0answers
32 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 ...
3
votes
2answers
41 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
54 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
39 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
30 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 $K$. ...