Questions related to the algebraic structure of algebraic integers

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8
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2answers
204 views

Elements of cyclotomic fields whose powers are rational

Suppose the polynomial $t^k - a$ has a root (hence splits) in $\mathbb{Q}(\zeta_k)$. For which $k$ does it follow that one of the roots of $t^k - a$ is rational? In particular, are there infinitely ...
0
votes
1answer
177 views

a calculation problem about haar measure

Here is a problem in charpter $2$ section $5$ in Algebraic Number Theory written by Jiirgen Neukirch. The problem is Let $K$ be a $p-adic$ number field, $v_p$ the normalized exponential ...
1
vote
1answer
105 views

what is the basis of neighbourhood of the element $1$ in a $p-adic$ field $K$?

There is a problem in Neukirch's Algebraic Number Theory,which is in Charpter $2$ section $5$ . The problem is : If $K$ is a $p-adic$ number field, then the groups $K^{*n}$, for $n$ belongs to ...
4
votes
1answer
446 views

A question on the topological structure of p-adic fields

This an exercise in Algebraic Number Theory written by Jürgen Neukirch. It is in chapter $2$, section $5$. The question is as follows: For a $\mathfrak{p}$-adic number field $K$, every subgroup of ...
8
votes
0answers
202 views

Is the splitting field unramified over a prime $\mathfrak{p}$ if the discriminant of the polynomial is coprime to $\mathfrak{p}$?

Is the following statement true? Let $R$ be a discrete valuation ring with quotient field $K$ and valuation $\nu$. Suppose that $f(x)\in K[x]$ is an irreducible separable polynomial with ...
3
votes
1answer
133 views

Find a finite extension of $\mathbb{Q}$ in which all primes split

Dear all, I would be grateful if someone could provide a solution to the following problem (using decomposition and inertia groups): Find a finite extension of $\mathbb{Q}$ in which all primes split. ...
7
votes
0answers
156 views

closure of units of number fields in the finite idele topology

Let $K$ be a number field. Denote by $\mathcal O _K^\times$ its rings of units and by $\mathcal O _{K,+} ^\times$ its ring of totally positive units. Further let us denote by $\mathbb A ...
31
votes
3answers
2k views

Sums of roots of unity

If the integral linear combination of some $n$th roots of unity has magnitude 1, does this necessarily imply that this linear combination is some root of unity as well? More precisely, let $\zeta_1, ...
5
votes
4answers
993 views

Lack of unique factorization of ideals

I'm aware of the result that integral domains admit unique factorization of ideals iff they are Dedekind domains. It's clear that $\mathbb{Z}[\sqrt{-3}]$ is not a Dedekind domain, as it is not ...
6
votes
2answers
253 views

Galois extension of $\mathbb{Q}_2$ with Galois group $(\mathbb{Z}/2\mathbb{Z})^3$

I am trying to solve this exercise: Prove that $\mathbb{Q}_2$ has a unique Galois extension F with Galois group $(\mathbb{Z}/2\mathbb{Z})^3$. Compute it's ramification groups. Here is what I have ...
5
votes
1answer
455 views

Prime splitting in an extension of prime degree

The following exercise and hint appear in Neukirch's Algebraic Number Theory (Section 9, Exercise 3, page 58) Let $L/K$ be a solvable extension of prime degree $p$ (not necessarily Galois). If ...
1
vote
0answers
182 views

Prime ideal splitting in field extension and its normal closure

The question is: Let L / K be a finite (not necessarily Galois) extension of algebraic number fields and N / K the normal closure of L / K. Show that a prime ideal p of K is totally split in L if and ...
5
votes
4answers
2k views

Splitting of prime ideals in algebraic extensions

I'm reading Algebraic Number Theory by Jurgen Neukirch. I have some problems with some of the exercises in Section 9 of Chapter 1. They are: 1) If $L / K$ is a Galois extension of algebraic number ...
2
votes
1answer
145 views

Difference between zeta sum and Euler product?

The fact that $$\sum_{n=1}^\infty \frac{1}{n^s} = \prod_{p}\frac{1}{1-p^{-s}}$$ is a consequence of unique factorization of primes. We could form a similar sum and a similar product of irreducibles ...
6
votes
1answer
751 views

Application of ideal class group?

I know what the class group is but I could not come up with any setting where we need to know what the class group actually is. Does anyone know of an example where we have two number fields with the ...
3
votes
0answers
732 views

Quick explanation: calculating class group of $\sqrt{-21}$

I'm trying to calculate the class group of $\mathbb{Q} (\sqrt{-21})$, and I've managed to confuse myself. I've used the standard Minkowski bound to show that we only need to consider principal ideals ...
4
votes
2answers
164 views

how to translate local reciprocity with galois groups into local reciprocity with weil groups?

I am trying to understand certain aspects of the Weil group $W_K$ for a $p$-adic field K, in particular how it does interplay with local class field theory. Let $L/K$ be a finite unramified ...
4
votes
3answers
716 views

Dedekind's theorem on the factorisation of rational primes

Let $K$ be an algebraic number field, and suppose its ring of integers is $\mathcal{O}_K = \mathbb{Z}[\theta]$ for some $\theta \in \mathcal{O}_K$. Let $f \in \mathbb{Z}[X]$ be the minimal polynomial ...
7
votes
1answer
1k views

What is the discriminant of a quadratic extension over a number field?

Let $K$ be a number field and $d \in \mathcal{O}_K \setminus \mathcal{O}_K^2$. What is the discriminant of the extension $K[\sqrt{d}]/K$ ? Do we know its ring of integers and which primes are split or ...
2
votes
0answers
105 views

Geometric understanding of principal/non-principal ideals

A number field $K$ with the $r$ embeddings into $\mathbb R$ and $2s$ pairs of conjugate embeddings into $\mathbb C$ can put into ring homomorphism with the product of rings $\mathbb R^r \times \mathbb ...
5
votes
1answer
254 views

Proving that isomorphic ideals are in the same ideal class

I've decided to study some number theory in my free time this summer, and have started reading Marcus' "Number Fields" and working through the exercises, but I've got myself stuck on one. This is the ...
2
votes
1answer
145 views

A question about local fields

I have a question which is probably caused by some confusion I have with extensions of local fields. Let us fix a finite extension of number fields $L/K$ and fix further $\mathfrak p$ denote a non ...
3
votes
1answer
151 views

The form $xy+5=a(x+y)$ and its solutions with $x,y$ prime

In another question I was asking if there are any different $x,y>2$ primes such that $xy+5=a(x+y)$. Where $a=2^r-1$, and $r>2$. I was thinking if it is able to find a Pell equation or a ...
4
votes
3answers
263 views

norm of an algebraic number with abs value smaller than 1

Let $\alpha \in \mathbb{C}$ be an algebraic number and $\alpha = \alpha_1, \alpha_2,...,\alpha_n$ its conjugates and $N(\alpha) = \prod_i \alpha_i$ its norm. Is it true that $|\alpha| < 1 ...
4
votes
1answer
118 views

Subgroups of index 3 in $1+p\mathbb{Z}_p$

Let $p$ be a prime. I'm trying to compute the subgroups of index $3$ in $\mathbb{Q}_p^\times$ to enumerate some cyclic extensions using CFT. I've essentially reduced the problem down to finding the ...
4
votes
1answer
194 views

About cyclic extensions of $\mathbb{Q}_p$

I'm trying to learn how to apply local class field theory and I thought about trying to enumerate some low degree abelian extensions of $\mathbb{Q}_p$. The easiest case is the quadratic extensions ...
10
votes
2answers
500 views

Cubic polynomials that generate the same extension?

For quadratic extensions we can easily determine when $\mathbb{Q}(\sqrt{a})=\mathbb{Q}(\sqrt{b})$ by checking if $a/b$ is a square and this is easy to prove. I was wondering if there are any good ...
9
votes
2answers
1k views

Ramification in a tower of extensions

I'm trying to make sense of all these theorems related to ramification. I was hoping someone would summarize these results. Assume we have: An extension $L/K$ and a some subextensions ...
3
votes
1answer
234 views

Gaussian Integer Cannot be Ordered

I just started to read Neukirch's Algebraic Number Theory book. On page 5 , the book has the following exercise. Show that the ring $\mathbb{Z}[i]$ cannot be ordered I don't quite understand ...
27
votes
1answer
1k views

Relation between the Dedekind Zeta Function and Quadratic Reciprocity

I was trying to learn a little about the Dedekind zeta function. The first place I looked at was obviously the Wikipedia article above. So my question comes from a sentence by the end of the article ...
3
votes
1answer
169 views

Silly question about a particular Frobenius automorphism

Let $K=\mathbb{Q}(\zeta_p)$. Then if $q\nmid p$ is any odd prime, we know that the Frobenius map $(q,K/\mathbb{Q})$ is just the map $(q,K/\mathbb{Q})(\zeta_p)=\zeta_p^q$. Now let ...
6
votes
3answers
1k views

Show that a specific ideal is not principal

In some cases, it is quite straightforward to prove that a specific ideal cannot be principal. For example, in the ring of integers of $\mathbb{Q}(\sqrt{-5})$, the ideal $(2,1+\sqrt{-5})$ is not ...
4
votes
1answer
175 views

Finiteness of class group in idelic language

How should I understand the compactness of $A_{\mathbb{K}}^1/\mathbb{K}^{\times}$ in classical non-idelic language? I suppose the notations are standard, but just for completeness, $K$ is a global ...
10
votes
2answers
304 views

Classifying splittings of primes?

I was wondering what general strategies are available to figure out if a prime splits? I know for quadratic extensions there aren't too many possibilities for how a prime can split, so we essentially ...
3
votes
4answers
919 views

Simple formula for integer polynomial with $2\sin(2\pi/n)$ as a root?

Is there a simple formula an integer polynomial that $2\sin(2\pi/n)$ satisfies? For $2\cos(2\pi/n)$ the answer is relatively nice. For any given $n$, we have $2\cos(2\pi/n)= z + z^{-1}$ where $z = ...
8
votes
4answers
540 views

Unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, and $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$

I want to prove the assertion: The unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ when $p \equiv 1 \pmod{4}$, respectively $\mathbb{Q}(\sqrt{-p})$ when $p \equiv 3 ...
4
votes
1answer
225 views

Total ramification $p = \epsilon \pi^n$ implies $F = K(\pi)$ and minimal polynomial of $\pi$ Eisenstein

Before I state my question, let me give the set-up / "what I know". Let $F / K$ be an extension of number fields of degree $n$. Let $v$ be a (discrete) valuation on $K$ and let $$ \mathfrak{O}_{v} ...
8
votes
2answers
456 views

Computing the Hilbert class field

Does anyone know any good source with nice examples of Hilbert class field computations? I'm trying to piece together the theory with some canonical examples.
3
votes
2answers
125 views

Step in Proof of Lemma in Narkiewicz _Elementary and Analytic Theory of Algebraic Numbers_

I was looking at the proof of Lemma 2.17 in Narkiewicz Elementary and Analytic Theory of Algebraic Numbers but don't understand a step. Let $p$ be a rational prime, $a$ be an algebraic integer of ...
1
vote
2answers
423 views

Hints for proving some algebraic integer rings are Euclidean

In my book - Algebraic Number Theory and Fermat's Last Theorem Ian Stewart, David Tall - there is the exercise: Prove that the ring of integers of $\mathbb{Q}(\zeta_5)$ (the 5'th cyclotomic ring) is ...
1
vote
1answer
161 views

What 'special' properties do real quadratic fields have?

Sorry for the vague title... I've proved a number theoretical result for the imaginary quadratic fields (it was already known for the rationals). I think it would be much easier to sell if I could ...
5
votes
1answer
803 views

How many real quadratic number fields have the class number 1?

I know that in general the number of ideal classes are not 1, and that there are only 9 imaginary quadratic number fields which are principal ideal domains, i.e. $\mathbb(Q(\sqrt{-m}))$ where m is 1, ...
1
vote
2answers
250 views

Where can I find Heegner's proof?

Where can I read a corrected up to date version of Heegner's solution of the class 1 problem of Gauss?
4
votes
1answer
155 views

Computing units of certain number fields

Some standard examples on various quals seems to be computing units/class numbers etc. of the ring $\mathbb{Q}(\alpha)$, where $\alpha$ is a root of either $X^3+aX+b$ or $X^5+aX+b$. My questions is ...
5
votes
1answer
421 views

Does every ideal class contains a prime ideal that splits?

Suppose you have a number field $L$, and a non-zero ideal $I$ of the ring of integers $O$ of $L$. Question part A: Is there prime ideal $\mathcal{P} \subseteq O$ in the ideal class of $I$ such that ...
7
votes
1answer
366 views

Any resource of the applications of the theory of class fields

We all agree that the theory of class fields plays an eminent role in modern number theory. Nevertheless, what was our main concern is that how to solve various Diophantine equations to which the ...
3
votes
2answers
293 views

Minimal polynomial

I have a cyclotomic field $\mathbb{Q}(\zeta_8)$, and want to know how I can find a minimal polynomial of this element $\zeta_8-2*\zeta_8^3$. Can we generalize this to any number field?
3
votes
0answers
106 views

Where can the original paper by Takagi in English be found?

As we all know, the theory of class fields began at the paper by Takagi on the abelian extensions of the field of rational numbers. Then one naturally has the following Where can one find the ...
10
votes
1answer
362 views

How to determine all valuations of the field $\mathbb{Q(\sqrt[n]{2})}$?

This is an exercise from the book Algebraic Number Theory by Jurgen Neukirch, on page 166. And, after solving several previous exercises, I found this to be particularly difficult to solve. I am ...
89
votes
1answer
7k views

What's the significance of Tate's thesis?

I've just sat through several lectures that proved most of the results in Tate's thesis: the self-duality of the adeles, the construction of "zeta functions" by integration, and the proof of the ...