Questions related to the algebraic structure of algebraic integers

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5
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1answer
34 views

$F[[T]] \times F[[1/T]]$, fundamental domain.

Let $p$ be a prime number. Here is a link which shows how to see that $$(\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T)))/\mathbb{F}_p[T, 1/T]$$is compact using an adelic result. (Here $\mathbb{F}_p[T, ...
6
votes
0answers
41 views

$y^2 = x^3 - 26$, exist ideal satisfying conditions?

For the solution $(x, y) = (3, 1)$ of $y^2 = x^3 - 26$, does there necessarily exist an ideal $I$ of the integer ring $\mathbb{Z}[\sqrt{-26}]$ of $\mathbb{Q}(\sqrt{-26})$ such that $(y + \sqrt{-26}) = ...
0
votes
3answers
34 views

How many integral ideals $\mathfrak{a}$ are there with the given norm $\mathfrak{N}(\mathfrak{a})=n$?

It is Exercise 1.6.1 in Jürgen Neukirch's number theory textbook(P38). I think the number $n$ here means $[K:\mathbb{Q}]$, the degree of field extention for algebraic number field K. I can image how ...
2
votes
1answer
19 views

Is a local field perfect?

Is any local field (with finite residue field) perfect (even with nonzero characteristics)? I know any finite field is perfect. I studied local fields by Neukirch's book.
5
votes
2answers
70 views

$L$-function, easiest way to see the following sum?

What is the easiest way to see that$$\sum_{(m, n) \in \mathbb{Z}^2 \setminus \{0, 0\}} (m^2 + n^2)^{-s} = 4\zeta(s)L(s, \chi)?$$Here $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to ...
6
votes
1answer
43 views

Quadratic field, $O_K/\mathfrak{p} = \mathbb{F}_p$, $O_K/pO_K$ is a finite field of order $p^2$.

Let $K$ be a quadratic field $\mathbb{Q}(\sqrt{m})$ where $m$ is a square free integer, and let $p$ be a prime number which does not divide $2m$. Where can I find a reference to a proof of the ...
7
votes
1answer
86 views

What is the Galois group of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$, where $\zeta$ is a $p^r$th root of unity?

Let $\zeta$ be a $p^r$-th root of unity and $\mathbb{Q}_p$ the p-adic numbers. I would like to know the Galois automorphisms of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$. I already know, the degree of ...
1
vote
1answer
17 views

Question in proof from James Milne's Algebraic Number Theory

I'm having difficulty understanding a step in a proof from J.S. Milne's Algebraic Number Theory (link). Here $\zeta$ is a $p$th root of unity and $\mathfrak p = (1-\zeta^i)$ for any $1\leq i\leq p-1$ ...
1
vote
1answer
32 views

extend trace and norm from a number field $K$ to $ K \otimes_{\mathbb Q} \mathbb R $

I read somewhere that we can extend the trace and the norm of a number field $K$ to the commutative algebra $ V=K \otimes_{\mathbb Q} \mathbb R$. Before state exactly my question, let me write the ...
6
votes
1answer
46 views

Cyclotomic polynomials, properties.

Let $F$ be a field of characteristic prime to $n$, and let $F^a$ be an algebraic closure of $F$. Let $\zeta$ be a primitive $n$th root of unity in $F^a$. I know that the monic polynomial $\Phi_n(X)$ ...
2
votes
1answer
49 views

p-th root does not become a p-th power when adjoined?

Suppose $k$ is a number field of characteristic zero, and $u$ is a unit of infinite order, which is not a $p$-th power in $k$. Show $\sqrt[p]{u}$ is not a $p$-th power in $k(\sqrt[p]{u})$. (You can ...
2
votes
1answer
19 views

Unit group of a field is divisible

In the lecture notes on Valuation theory, in Ex. $1.16$ on page $11$ we are asked to show that: If $k$ is an algebraically closed field, then $k^{\times}$ is a divisible abelian group. Isnt $k = ...
3
votes
0answers
26 views

The Frobenius Trace for an elliptic curve

Let E be an elliptic curve defined over $\mathbb{Q}$ (coeffs. there), and consider its $n-$torsion points in $\mathbb{C}$, $E(\mathbb{C})_{\text{tors}}[n]$. We know this group is isomorphic to ...
3
votes
1answer
63 views

Q($\sqrt[3]{2}$) - Unique Factorisation Domain?

I am considering the set of "integers" of the from $$ a+b\sqrt[3]{2} + c\sqrt[3]{4} $$ where $a,b,c$ are integers. It is easy to show this field is closed under addition and multiplication. I then ...
0
votes
2answers
66 views

What does Lang mean here by “the usual criterion”?

Let $K$ be a number field containing the $n$th roots of unity, $S$ a finite set of places containing all the archimedean ones and all those which divide $n$, $K_S$ the group of $S$-units of $K$, and ...
3
votes
0answers
51 views

Valuations on $\Bbb Q(t)$

Ex. $2.3.3$ in Algebraic Number Theory by Neukirch is the following: Let $k$ be a field and $K = k(t)$ the function field in one variable. Show that the valuations $v_{\mathfrak p} $ associated to ...
0
votes
1answer
9 views

$K_S$ modulo $K_S^n$, where $K_S$ is the group of $S$-units

Let $K$ be a field containing the $n$th roots of unity, $S$ a finite set of places containing all the archimedean ones, and $K_S$ the group of $S$-units, i.e. those $x \in K^{\ast}$ which are units at ...
1
vote
1answer
24 views

the field fixed by inertia group is the maximal unramified field

$K/F$ is a ablian galois extension of a number field. $\mathcal{O}_F$ (resp.$\mathcal{O}_K$) is the interger ring of $F$(resp.$K$). Let $\mathfrak{P}$ be a prime of $\mathcal{O}_K$ and ...
0
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0answers
24 views

square classes of quadratic extensions of 2-adic fialds

I have a question about square classes of quadratic extensions of 2-adic fields. I appreciate anybody help me to understand. Why all elements of $1+\mathfrak{p}^5$ are square in ...
5
votes
1answer
30 views

Why $R^q(\Gamma \circ \eta_{*}) (\Bbb G_{m, \eta}) = H^q(\eta_{ét}, \Bbb G_{m, \eta})$?

Let $X$ be a smooth, projective and connected curve over an algebraically closed field, and let $\eta \rightarrow X$ be its generic point (we also call the inclusion as $\eta$). I want to understand ...
2
votes
1answer
24 views

Bibilography: Riemann's hypothesis and positive semi-definite billinear forms

This is a bibliography request: I remember browsing through a book, some years ago, in a library, in which Riemann's hypothesis was proved over some type of fields (I cannot remember what type), the ...
3
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0answers
23 views

Why is this done? (Quadratic integer rings definition) [duplicate]

From Dummit & Foote pg. 229: Let $D$ be a squarefree integer. It is immediate from the addition and multiplication that the subset $\Bbb{Z}[\sqrt{D}] = \{a + b \sqrt{D} | a,b, \in \Bbb{Z}\}$ ...
6
votes
1answer
73 views

Integral basis of $\mathbb{Q}(\theta)$, where $\theta^3-\theta-4=0$

I am working on the text book "algebraic number theory" by Jurgen Neukirch(P15, exercise 6). To prove the integer basis is$ \{1, \theta, \frac{\theta^2+\theta}{2}\}$. After a long and tedious ...
13
votes
1answer
166 views

Sum of Reciprocals of Primes in Imaginary Quadratic Field Diverges (2014 Miklós Schweitzer)

Problem 5 of the 2014 Miklós Schweitzer states: Let $\alpha$ be a non-real algebraic integer of degree two, and let $P$ be the set of irreducible elements of the ring $\mathbb{Z}[\alpha]$. Prove that ...
-1
votes
1answer
74 views

If $\pi$ is not algebraic number then : is $\pi ^{n}$ algebraic number for $n >1$?

if $\pi$ is not algebraic number then : is $$\pi ^{n}$$ algebraic number for $n >1$ ? Thank you for any kind of help .
2
votes
1answer
20 views

Is a Galois group of a number field faithfully represented by its action on the set prime ideals of the ring of integers?

Is a Galois group $G$ of a number field $K$ faithfully represented by its action on the set prime ideals of the ring of integers $O_K$? This is true in some cases, like $Z[i]$. (Where we can see the ...
1
vote
1answer
34 views

List all elements in the residue field $Z[i]/(q)$

Consider a Gaußian prime $q$. How to list all elements in the residue field $Z[i]/(q)$? Is there any formulas or criteria? Here I'm looking for the case $q$ is a complex number, as I can do the real ...
0
votes
0answers
26 views

Discriminant ideal and short exact sequence of finite group schemes

Let $0 \to G' \to G \to G'' \to 0$ be a short exact sequence of finite flat commutative group schemes over a Dedekind domain $\mathcal O$ with field of fractions of characteristic zero. Let ...
1
vote
1answer
21 views

Number of solutions of $x_{0}x_{1} - x_{2}x_{3}$ over $\mathbb{F}_{p^{s}}$

I'm trying to solve the following exercise: Compute the zeta function of $x_{0}x_{1} - x_{2}x_{3}$ over $\mathbb{F}_{p}$. Well, for this, I need to find $N_{s}$, the number of solutions in the field ...
3
votes
0answers
34 views

Discriminant of $\mathbb{Z}[a,b]$

Let $K$ be an algebraic field extension of $\mathbb{Q}$. If $a\in \mathcal{O}_K$ is integral, then $Disc(\mathbb{Z}[a])=\prod_{i<j}( a_i - a_j)^2$ where the $a_i$ are the conjugates of $a$. Is ...
1
vote
1answer
14 views

moduli of the conjugates of a cyclotomic integer

I saw a theorem on the modulus of a cyclotomic integer: [Theorem] If every conjugate $\alpha '$of a cyclotomic integer $\alpha$ satisfies $|\alpha '|<\sqrt{2}$, then $\alpha$ is $0$ or a root of ...
0
votes
0answers
50 views

Non maximal prime ideals and localization

In the chapter on localization in Neukirch, Algebraic Number Theory, the following is mentioned: (A is an integral domain) Usually $S$ will be the complement of a union $\bigcup_{\mathfrak p \in ...
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0answers
28 views

Ramification index in a number field extension

I've a basic question: Let $L/K$ be an extension of number fields, $\mathfrak p$ a prime ideal of $\mathcal O_K$ and let $\mathfrak p \mathcal O_L=\prod_i P_i^{e_i}$ where $P_i$ are primes in $L$ and ...
2
votes
0answers
34 views

Good examples of (families of) tamely ramified extensions?

I am looking for examples of families of tamely ramified extensions of $\mathbb{Q}$. Thanks.
6
votes
1answer
43 views

How to geometrically interpret intertia of primes in field extensions?

I am trying to understand the intuition of thinking about number theoretic ideas in terms of geometric ones. For example, ramification is something that happens when a "covering" space of a Riemann ...
2
votes
1answer
31 views

Galois of successive polynomials in the series expansion $e^x=\sum_{n=0}^\infty\frac{x^n}{n!}$.

I read in a French paper an assertion without proof that I had not known before and that really catches my attention: The Galois group over $\mathbb{Q}$ of the equation ...
2
votes
3answers
57 views

Understanding $p$-adic fields

OK, I'm completely lost on this. Define the $p$-adic integers $\mathbb{Z}_p$ as the projective limit $$\lim_{\leftarrow} \mathbb{Z}/p^n \mathbb{Z}.$$ So, if $a \in \mathbb{Z}_p$, then $a$ can be ...
5
votes
1answer
36 views

Is every non-archimedean absolute value on a number field equivalent to a $|\cdot|_{\mathfrak{p}}$?

Let $K$ be an algebraic number field, i.e. a finite field extension of $\Bbb{Q}$. I would like to prove that every non-archimedean absolute value on $K$ is equivalent to $$ |x|_{\mathfrak{p}} := ...
1
vote
1answer
18 views

Does the field norm commute with field morphisms?

Consider a field extension $K$ of $\Bbb{Q}$, a finite extension $L$ of $\Bbb{Q}_p$ for some fixed prime number $p$, and a field morphism $\sigma \colon K \to L$ such that the diagram $\require{AMScd}$ ...
3
votes
1answer
38 views

Elliptic curves, reduction map, $E_n$

Let $E$ be the elliptic curve and set $\phi: E(\mathbb{Q}_p) \rightarrow E(\mathbb{F}_p)$ to be the reduction morphism. Define $E_n := \{(x:y:z) \in \ker \phi | x/y \in p^n\mathbb{Z}_p\}$. I'm busy ...
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0answers
20 views

Is it possible to describe the places of $K$ purely in terms of the algebra of $\mathcal{O}_K$?

By "purely in terms of the algebra" of $\mathcal{O}_K$, I mean without any reference to the real numbers, topological fields, etc. Of course we can get the finite places by considering ...
2
votes
0answers
18 views

Non-archimedean exponential valuation and integral closure

I am trying to solve the following problem from Neukirch's book on ANT: Let $L|K$ be a finite field extension, $v$ a nonarchimedean exponential valuation, and $w$ an extension to $L.$ If ...
0
votes
0answers
14 views

Conductor in a number field is non-zero [duplicate]

Let $L|K$ be an extension of number fields and define $\mathfrak J = \{\alpha \in \mathcal O_L\mid\alpha\mathcal O_L \subset \mathcal O_K[\theta]\}$, the largest ideal of $\mathcal O_L$ contained in ...
1
vote
1answer
39 views

Descending chain of ideals becoming stationary

Exercise 3.7 of Algebraic Number Theory (Neukrich) is: In a noetherian ring R in which every prime ideal is maximal, each descending chian of ideals $\mathfrak{a_1 \supset a_2 \dots}$ becomes ...
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0answers
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Proving factorization into prime ideals in a Dedekind domain

Let $\mathcal O_K$ be a Dedekind domain and $\mathfrak p$ a prime ideal in $\mathcal O_K$. Assume that we have shown the existence of a fractional ideal $\mathfrak p^{-1}$ such that ...
2
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0answers
19 views

What is binary norm of quadratic fields of sum of two squares such that one of them is necessarily even like $a^2 +4b^2?$

I am trying to simplify an expression which I have reached, suppose a number can be represented in the form of $D=a^2 + 4b^2$. What is binary norm of $D$, or how else can it be represented?
2
votes
1answer
32 views

How to express Witt vectors by their ghost components

Let $x_1,x_2,\cdots$ be infinitely many indeterminates, the ghost components of the Witt vector $x=(x_1,x_2,\cdots)$ is given by \begin{equation*} x^{(n)}:=\sum_{d\mid n}dx_d^{n/d}. \end{equation*} ...
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0answers
29 views

Is this polynomial irreducible over the rationals?

Prove (or disprove): Define $T_n(x)$ as the Chebyshev polynomial of the first kind with degree $n$. If $p$ is an odd prime, then $\sqrt{\frac{T_p(x)-1}{x-1}}$ is an irreducible polynomial over the ...
1
vote
1answer
40 views

Class number and complex conjugation

Let $h$ be the be the class number of the ring of integers of the $p$th cyclotomic field. Suppose $p\mid h$ and let $I$ be an ideal of order $m$ such that $p \mid m$. Does $p$ divide the order of $I ...
0
votes
1answer
16 views

Why is there a set $W$ (to be described below) such that $\mathbb{A}_K = W + K$?

To prove the compactness of $\mathbb{A}_{\mathbb{Q}}/ \mathbb{Q}$ (and hence $\mathbb{A}_K/K$ for an arbitrary number field $K$), one finds a set $W \subseteq \mathbb{A}_{\mathbb{Q}}$ of the form $$ ...