Questions related to the algebraic structure of algebraic integers

learn more… | top users | synonyms

1
vote
1answer
37 views

$p^k$-cyclotomic polynomials

Let $p$ be a prime integer. Let $\Phi_p(X):=X^{p-1}+X^{p-2}+\ldots+X+1$ and for any positive integer $k$ let $\Phi_{p^k}(X):=\Phi_p(X^{p^{k-1}})$. I'm asked to show that every coefficient (but the ...
2
votes
1answer
81 views

How deep is the connection between $\mathcal{O}$ for a structure sheaf and $\mathcal{O}$ for a ring of algebraic integers?

So clearly there's some connection because in both contexts you have a ring and its function field involved; in one setting that's $\mathcal{O}_X(U)$ and $\mathcal{O}_{X, \eta}$ and in the other it's ...
1
vote
1answer
67 views

Computing ring class field from ray class field

Let $K=\mathbb Q(\sqrt{-d})$ be any imaginary quadratic field. Let $O_{\mathcal{K}}$ be its maximal order and $O$ be any order. Let $m$ be the conductor of $K$. Is it possible to compute ring class ...
2
votes
1answer
72 views

Proof involving characters

I'm self studying from a Classical Introduction to Modern Number Theory by Ireleand and Rosen. In the exercises for the chapter on Gauss and Jacobi sums I came across this question. Let $\chi$ be a ...
2
votes
1answer
35 views

This particular element of the inertia may depends on these prime ideals?

Currently I'm reading an article. But since I'm not very experienced, I ended up getting stuck on a problem with the following lemma: $``$Let $K/\mathbb{Q}$ be a number filed, let $\mathfrak p$ be a ...
3
votes
2answers
99 views

Ideals generated by roots of polynomials

Let $\alpha$ be a root of $x^3-2x+6$. Let $K=\mathbb{Q}[\alpha]$ and let denote by $\mathscr{O}_K$ the number ring of $K$. Now consider the ideal generated by $(4,\alpha^2,2\alpha,\alpha -3)$ in ...
1
vote
1answer
310 views

Ramification index in number fields extension

Let $K$ and $L$ be number fields, namely $$K:=\mathbb{Q}[\sqrt{pq}]\ \ \ \textrm{and}\ \ \ L:=\mathbb{Q}[\sqrt{p},\sqrt{q}]$$ where $p,q$ are rational prime numbers, with $p\equiv 1\pmod{4}$ and ...
3
votes
0answers
129 views

How does Dedekind's Theorem work for a prime dividing the discriminant of a number field?

Let $f \in \mathbb{Z}[x]$ be an irreducible monic polynomial, let $N$ be its splitting field, and let $G$ be the Galois group of the extension $N/\mathbb{Q}$. Let $p$ be a prime dividing the ...
2
votes
2answers
43 views

elements with bounded conjugate in a number field

Suppose I have a number field $K$ of degree $n$, and $O_K$ its algebraic integers. Could anyone explain me how to show that the following set $S$ $$ S = \{ x \in O_K : |x^{(i)}| < B,\ for \ each \ ...
4
votes
1answer
226 views

Rational Points on Elliptic Curves

I have this homework problem: Can there be an elliptic curve, view as a projective curve, with no rational points with at least one 0 as a coordinate?
1
vote
1answer
161 views

Showing Quotient Ring is a Field

Consider the ring $S=\mathbb{Z}[\alpha]$, where $\alpha = \sqrt[3]{2}$, and ideal $I=(5,\alpha^{2}+3\alpha -1)$. I wish to show that $S/I$ is a field of order 25. Any solutions/suggestions? I would ...
3
votes
1answer
557 views

Generalized Euler phi function

Let $n$ be an integer. There is a well-known formula for $\phi (n)$, where $\phi$ is the Euler phi function (totient). Essentially, $\phi(n)$ gives the number of invertible elements in ...
5
votes
1answer
54 views

How to relate the valuation of x/y (For a minimal Weierstrass equation)

I'm reading an article about elliptic curves, but since I'm not very experienced on this subject, I ended up getting stuck. The problem starts as: "Let $K/\mathbb{Q}$ be a number field and $E/K$ an ...
4
votes
1answer
440 views

Ray class field - Ring class field

Let $K=Q(\sqrt{-d})$ be an imaginary quadratic field. Let $O_{K}$ be its maximal order and $O$ be any order of $K$. Let $m$ be the conductor. can the ray class field and ring class field be same?
7
votes
0answers
95 views

Ex. $2.30$ in Silverman Adv. Topics

I would like to refer you to Exercise $2.30(c)$ in Silverman's Advanced Topics in Elliptic Curves. Question: Let $E/L$ be an EC with CM by $K$. Assume that $K\nsubseteq L$, and let $L'=LK$ and let ...
1
vote
1answer
79 views

Ring class field of a number field K

Let $K=Q(\sqrt{-1})$. Let $m$ be the conductor of $K$ and $m=3$, then $Cl_{m}(K)=2$. Now how to compute the ring class field of $K$
0
votes
2answers
68 views

When $a$ is even, the difference between $(a/2) \mod N$ and $(a \mod N)/2$?

folks. Could I ask for your help? Let $N$ be a positive integer and $a$ be an even integer, i.e., $a=2x$ for an integer $x$. Then think of $W_N^{\frac{a}{2}}$, where $W_N=e^{j\frac{2\pi}{N}}$. ...
1
vote
1answer
71 views

Mathematica code as reference?

I am preparing a paper wherethere are a lot of algebraic simplifications needed. One of them takes 3 pages to get the result. Instead of consuming all the pages can I cite a mathematica code which is ...
4
votes
0answers
117 views

Why does there exist a totally positive element?

Given a totally real number field $K_0$ and its totally imaginary quadratic extension $K$. Does there exist an element $\psi\in O_K$ such that $-\psi^2$ is totally positive in $K_0$? Why?
1
vote
1answer
109 views

Field norm defined with or without absolute value?

I'm studying about the valuation for Euclidean Domains and quadratic fields $\mathbb{Q}(\sqrt{\theta})=\{ \alpha: \alpha=a+b\sqrt{\theta}, a,b \in \mathbb{Q} \}$ and I'm not sure whether we start with ...
1
vote
2answers
99 views

Rings with finitely many prime ideals.

What follows is an argument i found in my textbook which i can't understand. Let $R$ be a Dedekind domain, with quotient field $K$, $R'$ is the integral closure of $R$ in a finite and separable ...
7
votes
1answer
382 views

Ring class field of $\mathbb{Q}(\sqrt{-19})$

I am looking an explicit form for the ring class field of the order $\mathbb{Z}[\sqrt{-19}]$ in the quadratic field $\mathbb{Q}(\sqrt{-19})$. Does anyone know if there is some and how it is?
5
votes
1answer
95 views

Equivalent conditions for the ramification group(s) in a finite Galois extension

Let $L|K$ be a finite Galois extension and $v$ a discrete normalized valuation on $L$ such that its restriction to $K$ extends uniquely to $L$. (1) Why is $G_1=\{\sigma\in G(L|K)\mid ...
2
votes
3answers
149 views

Ring of integers in a cubic extension

Let $L=\mathbb{Q}[\alpha]$, with $\alpha^3=10$. How can be proved that $$\frac{\alpha^2+\alpha+1}{3}$$ is in $O_L$, the ring of integers of $L$?
8
votes
1answer
206 views

Quadratic extensions of $\mathbb Q$

This is a question from Lang's ANT, Thm 6, ch.IV, $\S2$. It states that every quadratic extension of $\mathbb{Q}$ is contained in a cyclotomic extension and that it's a direct consequence of the ...
2
votes
1answer
47 views

If $x^2+ax+b$ is an integer for every integer $x$ then comment on the coefficients $a$ and $b$ MCQ

Probably a more general category (number theory) multiple choice question but no clue how to get to a clear conclusion . Here's how it goes : Q) If $x^2+ax+b$ is an integer for every integer x ...
3
votes
1answer
81 views

Regarding valuation on function field

Notations: Let $k$ be a field of characteristic $\neq 2$, $k(X)$ be a function field. Suppose $p(X)\in k[X]$ is an irreducible polynomial and $\alpha$ be a root of $p(X)$. Let ...
6
votes
1answer
171 views

Eigenvalues of Multiplication in algebraic number field

Finally I have a new question which is puzzling me. I am currently reading a manuscript so there is no reference or link I can provide. I will try to put all the information needed in here. This ...
1
vote
1answer
37 views

Number fields containing the values of a character

This is a basic question- Let $K$ be an algebraic number field, $\Gamma$ a finite group, and $R(\Gamma)$ the ring of virtual characters of $\Gamma$ with values in the algebraic closure $\mathbb{Q}^c$ ...
2
votes
2answers
187 views

Ideal norm in a quadratic field

Let $K=\mathbb{Q}[\sqrt{d}]$ be a quadratic field with discriminant $d_K$, let $\mathfrak{a}=(a,\frac{b-\sqrt{d_K}}{2})$ be an ideal. Does the norm $N(\mathfrak{a})=a$? How to prove it?
1
vote
1answer
151 views

Isomorphism of the ideal class group with a cyclic group

Let $K=\mathbb{Q}(\sqrt{-17})$. Show that the ideal class group $Cl_K$ is isomorphic to $\mathbb{Z}/4\mathbb{Z}$. We know that the class number is 4...How i show that $Cl_K$ is cyclic?
2
votes
1answer
86 views

Group of finite ideles

A simple question: If $\overline{\mathbb{Q}}$ denotes the algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$ then what is the definition of the group of finite ideles of $\overline{\mathbb{Q}}$? ...
4
votes
2answers
285 views

Roots of monic polynomial over a number ring

If $R$ is a number ring with number field $K$ and $f$ is a monic polynomial over $R$, then I want to show that any root of $f$ is an algebraic integer.
0
votes
1answer
100 views

Square of an algebraic integer is an algebraic integer

For some $\alpha\in\mathbb C$ let $E=\mathbb Q(\alpha)$ be a number field and $\mathcal I$ its ring of integers. Suppose $(a_1+b_1\alpha)\cdots(a_k+b_k\alpha)=\beta^2$ for some $\beta\in E$ and ...
7
votes
1answer
288 views

Let $F_n$ be a Fibonacci number and $p$ a prime. Verify that for $p \le 61$, if $p\equiv\pm1 \pmod{5}$ then $p\mid F_{p-1}$

Define the Fibonacci entry point of $p$ to be the least integer $n$ such that $p\mid F_n$ So for example, for $p = 3$ - the Fibonacci entry point is $n = 4$ since $F_4 = 3$ and obviously $3\mid 3$. ...
5
votes
1answer
186 views

$J(\mathcal{O})\cong\bigoplus_{\mathfrak{p}} P(\mathcal{O}_\mathfrak{p})$ for one-dimensional Noetherian domains (from Neukirch)

I'm having trouble in understanding Neukirch's proof of the proposition in the title ((12.6), p. 75 of his Algebraic Number Theory book). He uses the following CRT-like proposition: If ...
0
votes
2answers
93 views

Primes corresponding to embeddings of a number field

Let $k$ be a number field. Define a prime of $k$ to be an equivalence class of absolute values on $k$. If $\sigma:k\hookrightarrow \mathbb{C}$ is an embedding of $k$ into the complex numbers then we ...
20
votes
1answer
707 views

Why did Gauss think the reciprocity law so important in number theory?

Gauss's Disquitiones Arithmeticae centers around the quadratic reciprocity law. It seems that he developed the genus theory of integral binary quadratic forms to find a natural proof of the quadratic ...
2
votes
1answer
80 views

Integral elements of a non-archimedean completion of a number field

Let $k$ be a number field, $v$ a discrete non-archimedean valuation on $k$. Let $k_v$ be the completion of $k$ wrt $v$ and $\mathcal O_v$ its valuation ring. My question is: If $x\in k_v$ then is ...
6
votes
1answer
128 views

Ramification in the ring of all algebraic integers

If $F$ is a finite extension of $\mathbb{Q}$ then its of integers $R$ is a Dedekind domain, and has unique factorization of ideals into powers of prime ideals. For each prime number $\ell$, you can ...
1
vote
1answer
219 views

A non-Archimedean norm definition can be strengthened.

See Andrew Baker's p-adic notes: For a non-Archimedean norm $N$ it is true that "$N(x + y) \leq \max\{N(x), N(y)\}$, with equality if $N(x) \neq N(y)$." Having trouble proving this. Please ...
5
votes
1answer
111 views

How the ring of algebraic numbers looks like?

Suppose I have an algebraic number field $K = \mathbb Q(\alpha)$ for some $\alpha \in O_K$, ring of algebraic integers. Is there a criterion that tells us when $O_K =\mathbb Z[\alpha]$ by any ...
1
vote
1answer
75 views

Ireland-Rosen Hecke Character for $y^2=x^3-Dx$

I would like to refer you to page $310$ of Ireland-Rosen: A Classical Intro to Modern Number Theory. Firstly, to construct the Hecke character, it is enough to specify $\chi(P)$ for prime ideals $P$ ...
1
vote
1answer
89 views

Quadratic equation family with largest real root in Cyclotomic extension

Let the $\alpha_{k}$ be the largest real root by absolute value of $ 2x^2-2kx-(k-1)=0$ for all $k\ge1$. For what values of $k$ does $\alpha_{k}$ sit in a cyclotomic extension? How does one explicitly ...
5
votes
1answer
117 views

Element of with n0n-zero trace

Let $F$ be a field of characteristic $p$ and $K$ a finite, separable extension of $F$ such that $p \mid [K : F]$. I want to show that there must exist an element of $K$ with non-zero trace. One idea ...
5
votes
1answer
156 views

The uniqueness of a special maximal ideal factorization

The following problem is from Michael Artin's Algebra, chapter 12, M.6, unstarred: Let $R$ be a domain, and let $I$ be an ideal that is a product of distinct maximal ideals in two ways, say ...
1
vote
1answer
45 views

Quotients of a valuation ring in the completion of a number field

Let $k$ be a number field, $v$ a discrete (non-archimedean) valuation on $k$. Let $k_v$ be the completion of $k$ with respect to $v$. Also let $\mathcal{O}_v$ be the valuation ring of $k_v$ and ...
3
votes
1answer
73 views

powers of a unit in a number field never in a subfield

By using the ranks of the unit groups, one can see that if $K$ is not a CM field, then the unit group of $K$ contains a unit not in any subfield. However, is it also possible to find a unit, ...
10
votes
1answer
328 views

Primes of ramification index 1 with inseparable residue field extension

I've been reading through Neukirch's Algebraic Number Theory, and I'm a little puzzled about a possibility with ramification of primes. As usual, let $\mathcal{O}_K$ be a Dedekind domain with field ...
2
votes
1answer
134 views

$p$ splits completely in $\mathbb{Q}(\zeta_l)$ implies it splits in $\mathbb{Q}(\sqrt{-l})$

Let $l>3$ be a prime such that $l\equiv 3 (\textrm{mod } 4)$. Let $p$ be an odd prime such that $p\equiv 1 (\textrm{mod } l)$. Then we can prove directly that $p$ splits in $\mathbb{Q}(\sqrt{-l})$. ...