Questions related to the algebraic structure of algebraic integers

learn more… | top users | synonyms

2
votes
1answer
86 views

Define, $p^{-1} = \{x \in K: xp \subset D\}$. Then show that there exists a non zero $c \in D$ such that $cp^{-1} \subset D$.

Let $D$ be an integral domain and $K$ be its field of fraction. Also, given that $D$ is Notherian, Integrally closed, and every non-zero prime ideal in $D$ is maximal ideal. Let $p$ be a ideal of ...
2
votes
1answer
20 views

Relationship between index of a basis for a ring of integers and the integer span of the basis

In my notes for number theory, after having proved the following: Corollary: For a number field $K$, if $x_1,\dots,x_n\in\mathcal{O}_K$ is a basis for $\mathcal{O}_K$ over $\mathbb{Q}$ and ...
1
vote
2answers
40 views

Linearly independent algebraic integers form an integral basis

Let $K$ be a number field and let $\alpha_1,\alpha_2,...,\alpha_n$ be a set of linearly independent algebraic integers such that $$\Delta_{K/\mathbb{Q}}(\alpha_1,\alpha_2,...,\alpha_n)=d$$ where $d$ ...
3
votes
0answers
45 views

$P_{K,1}(\mathfrak m)\subset \operatorname {ker} \Phi_{\mathfrak m,L|K} \subset \operatorname {ker} \Phi_{\mathfrak m,M|K}$ imples $M \subset L$

Let $K$ be a number field and $L, M$ finite abelian extensions. Let $\mathfrak m$ be a modulus. Consider the two Artin maps $ \Phi_{\mathfrak m,L|K}$ and $ \Phi_{\mathfrak m,M|K}$. Let ...
6
votes
0answers
133 views

Adelic/Idelic method for number fields

I'm searching for a place where is developed all the machinery of adeles and ideles of a number field; the functional equation for the zeta functions is gained by idelic integration, and it's seen the ...
2
votes
0answers
32 views

Confusing application of power residue reciprocity in Milne's CFT

Hey I am trying to figure out the details of the proof of Theorem 5.14 (p.246) in Milne's CFT (see here). I hope somebody is familiar with this. But let me sketch the proof and what I don't ...
4
votes
0answers
51 views

Prime number theorem for Dedekind domains

Let $\mathscr P\subseteq \mathbb N$ be the set of prime numbers. The prime number theorem tells us that if $\pi(x)=|\{p\in\mathscr P\colon p\leq x\}|$ then $\pi(x)\sim \frac{x}{\log x}$. Now one could ...
4
votes
1answer
260 views

Forcing the discriminant of an integral basis to be a Carmichael number.

I was thinking about the following lemma recently. Lemma: Let $K=\mathbb{Q}(\theta)$ for some algebraic number $\theta$ and let $n=[K:\mathbb{Q}]$. If $\{\tau_1, \,\dots\,, \tau_n\}$ consists of ...
0
votes
1answer
73 views

What does it mean for a “place” to divide?

A proof I'm trying to understand refers to the set of all finite places dividing an algebraic integer x. What does this mean? I can't seem to find a definition in any of the texts I've looked at. ...
5
votes
1answer
72 views

A problem in Galois Theory

While reading algebraic number theory, I came across the following statement: Let $K$ be a galois extension over $\mathbb{Q}$ and $H$ be the Hilbert class field (maximal unramified abelian extension) ...
2
votes
1answer
77 views

from Ireland and Rosen: when a prime remains inertial

I'm reading Ireland and Rosen's number theory book, and i'm having trouble with proposition 13.1.3 ii): Let F be $\mathbb{Q(\sqrt d)}$ where $d$ is a square free integer,and $p$ and odd prime, and ...
2
votes
2answers
28 views

How to show convergence with respect to $\left|\cdot\right|_{p}$?

I am having a difficulty showing convergence or divergence with respect to $\left|\cdot\right|_{p}:\mathbb{Q}\to\mathbb{Z}\cup\left\{ \infty\right\}$ where $a=p^{n}\frac{a'}{b'}\to p^{-n}$ . I ...
3
votes
1answer
40 views

$p$-th roots of unity adjoined to a $\mathfrak{p}$-adic field

I want to prove the following: Let $k$ be a number field and $S$ a set of primes of $k$ containing the primes $S_p$ that lie over the rational prime $p$. Then the extension of $k$ by the group of ...
17
votes
2answers
654 views

Prove that $x^3 + y^3 = z^3$ has no integer solutions as simply as possible

Can someone prove the special case of Fermat's Last Theorem for $n=3$, i.e., that $$x^3 + y^3 = z^3,$$ has no positive integer solutions, as simply as possible? I have seen some good proofs, but ...
6
votes
1answer
129 views

How to arrive at Ramanujan nested radical identity

I've come across a very curious Ramanujan identity $$\sqrt[6]{7 \sqrt[3]{20} - 19} = \sqrt[3]{\frac{5}{3}} - \sqrt[3]{\frac{2}{3}}$$ You could probably prove this by taking the 6th power of both ...
1
vote
1answer
41 views

Find index of $\mathbb{Z}[\theta]$ in $\mathbb{Z}_{k}$

Find index of $\mathbb{Z}[\theta]$ in $\mathbb{Z}_{k}$ i.e $[ \mathbb{Z}_{k}:\mathbb{Z}[\theta]]$, where $K=\mathbb{Q}(\sqrt{5}),\theta=\sqrt{5}$ As $5\equiv 1 \pmod 4$ the ring of integers of $K$ ...
1
vote
1answer
81 views

Field of all algebraic reals over $\mathbb{Q}$ has infinite order.

I am trying to show that field of all algebraic reals over $\mathbb{Q}$ has infinite degree. I guess that $$1,\sqrt{2},\sqrt[3]{2}, \sqrt[4]{2}, ...$$ are lineary independent but can't prove it.
1
vote
0answers
19 views

Class groups containing only elements of odd order

I need to find an infinite number of real quadratic fields such that their class groups contain only elements of odd order. Is it true for all real quadratic fields with prime discriminant? Can we ...
2
votes
3answers
62 views

Legendre symbol question for infinitely many primes of form 4k + 3

Given a positive integer n, how would one show that there are infinitely many primes p of the form 4k + 3 that have Legendre symbol (n/p) = -1? From the comments I have received thus far, it has been ...
3
votes
1answer
55 views

Example 4.3.19 in Liu: unramification with schemes and numbers

In exemple 4.3.19 of Liu's book one hase $L/K$ an extension of number fields with integer rings $\mathcal{O}_L$ and $\mathcal{O}_K$, $\mathfrak{q}\subseteq\mathcal{O}_L$ a prime ideal and ...
3
votes
1answer
69 views

Division Theorem in the integers adjoin square root of minus two

Q. let $\mathbb{Z}[\sqrt{-2}] = \{a + b\sqrt{-2} \mid a,b \in \mathbb{Z} \}$ Show that if $s,t \in \mathbb{Z}[\sqrt{-2}],$ $t\not = 0$ then there exists $r,q \in \mathbb{Z}[\sqrt{-2}]$ s.t.$ s = tq + ...
2
votes
1answer
51 views

$\mathbb{Z}/p\mathbb{Z}$ extension of a local field

Let $K/\mathbb{Q}_l$ be a finite extension. How can one prove that the number of extensions of $L/K$ such that $Gal(L/K) \cong \mathbb{Z}/p\mathbb{Z}$ is finite. If i'm not mistaken class field ...
0
votes
0answers
27 views

What is a locally abelian extension of number fields?

I recently came across the following term in a number theory paper: a locally abelian extension of number fields. Would anyone know what it means as I haven't come across it before? Many thanks. ...
4
votes
0answers
115 views

Difficulty of exercises in several books

Since I am teaching myself, I have no idea of the difficulties of exercises in many books. I want to know whether I am not understanding the content of these books if I cannot easily solve the ...
4
votes
2answers
116 views

The ring of integers of $\mathbf{Q}[i]$

Is there a relatively "simple" (in the sense that it does not require knowledge of algebraic number theory) proof that the ring of integers of the algebraic number field $\mathbf{Q}[i]$ is ...
3
votes
1answer
51 views

Correspondence between prime ideals and Galois orbits of affine points on an elliptic curve

In notes of prof. W.Stein - http://wstein.org/edu/2010/581b/stein-algebraic_number_theory.pdf - the first paragraph of page 112 has the following told: "When $K$ is a perfect field, the prime ideals ...
2
votes
1answer
72 views

factorization of ideals

Let $L$ and $K$ be number fields such that $L/K$ is a finite extension. Suppose $\mathfrak{a},\mathfrak{b}$ are ideals in $\mathcal{O}_K$ and $\mathfrak{a}\mathcal{O}_L|\mathfrak{b}\mathcal{O}_L$. ...
5
votes
2answers
181 views

Determining ring of integers for $\mathbb{Q}[\sqrt{17}]$

I'm trying to find the ring of integers of $\mathbb{Q}[\sqrt{17}]$, and it comes down to determining the set $\{(a,b)\in\mathbb{Q}^2\mid 2a\in \mathbb{Z}, a^2-17b^2\in\mathbb{Z}\}$. How can I ...
2
votes
0answers
51 views

quadratic rings of integers vs cubic rings of integers in number fields

I would appreciate if someone could give me some clues about cubic $\mathbb{Z}$-rings of number fields. So far I have only learned about quadratic rings and I would like to see if there are any ...
4
votes
1answer
44 views

Properties of Dedekind zeta function

Suppose $K$ is a quadratic field and $a_K(n)$ denotes the number of ideals in the ring of integers of $K$ whose norm is equal to $n$. Then I need to show that $$\sum_{n\leq x} a_K(n)=O(x).$$ Clearly ...
3
votes
0answers
76 views

An application of Hensel's lemma to some different fields

I would like to prove the following. Let $p>2$ be a prime number, $\mathbb{Q}_{p}$ the field of p-adic numbers Let $u\in \mathbb{Z}_{p}^{\times}$ be a unit. (1)Prove that the following are ...
0
votes
0answers
18 views

Sum of roots of unity bounded away from 0

Let $n\in \mathbb N$ and $\zeta_N$ be a primitive $N$th root of unity. Let $a_k\in \mathbb Z,0\le k<N$. Assuming that the sum is nonzero, find a lower bound on the absolute value of $$ ...
2
votes
2answers
78 views

$p$-adic valuation.

Let $\alpha_1,\alpha_2\in \mathbb Z_p$ such that $v_p(\alpha_1)<v_p(\alpha_2).$ How to prouve that $v_p(\alpha_2-\alpha_1)=v_p(\alpha_1)$ ? I think this is a stupid question but I'm really ...
3
votes
1answer
57 views

Nilpotent action on $p$-group

Let $A$ be a finite, abelian $p$-group and $\Gamma$ is a multiplicative topological group isomorphic with the additive group of $p$−adic integers $\mathbb Z_p.$ and let $\gamma_0$ a topological ...
1
vote
1answer
34 views

If $g$ is a primitive $p$-root of unity, then $g^n\equiv 1 \pmod p \iff (p-1)|n$

This statement is from Caratheodory's Theory of Functions of a complex variable vol.1 p.282: Let $s_n(p):=0^n+1^n+2^n+...(p-1)^n$. Then if $p$ is a prime and $g$ is any number that doesn't divide ...
1
vote
0answers
19 views

Intersections of all open subsets of finite index in the idele group

Let $K$ be a number field. Why is the intersections of all open subgroups of finite index in the idele group $\mathbb I_K$ equal to $\overline{K^{\times}(K^{\times}_{\infty})^0}$? Also, I'm having a ...
1
vote
1answer
77 views

Being Galois stable under completion?

Let $R$ a Dedekind Domain, $K = \mathrm{Frac}(R)$ the fraction field, $L/K$ a finite galois extension, $R'$ the integral closure of $R$ in $L$. Then $R'$ is Dedekind again. Let $\mathfrak{p} \subset ...
1
vote
0answers
40 views

Question about $p$-adic exponential

Let $p$ be a prime number, and $K$ a finite extension of $\mathbb Q_p$ and $S=p^N\mathcal O_K$ where $\mathcal O_K$ the ring of integers of $K.$ I know that for $N>>0$ enough large the $p$-adic ...
5
votes
1answer
129 views

What is the best book learn Galois Theory if I am planning to do number theory in future?

What is the best book learn Galois Theory if I am planning to do number theory in future? In a year i'll be joining for my Phd and my area of interest is number theory. So I want to know if there is ...
3
votes
1answer
55 views

Weird definition of norm's triangle inequality

Cassels (Froehlich & Cassels, ANT) uses a rather unusual definition of triangle inequality when he defines a norm. He states that a norm should satisfy the following (in lieu of ...
1
vote
0answers
34 views

Definition of tame ramification from the ramification groups

Let $R$ be a d.v.r., $K$ its field of fractions, $L/K$ a separable finite field extension and $S$ the integral closure of $R$ in $L$. Let $p$ be the prime ideal of $R$ and assume that $K$ is complete ...
0
votes
0answers
15 views

Quotients groups in the higher ramification sequence

I have a problem in the proof (which is left to the reader) of Proposition (10.2) of Chapter II of Neukirch's book Algebraic Number Theory. Here it is. For any valued field $K$, I write $v_K$ the ...
3
votes
1answer
49 views

Euler's formula for trace of $x^i/g'(x)$

Let $L/K$ be a finite separable extension of fields generated by an element $x$ with minimal polynomial $g(X)=0$. Assume that $K$ is the fraction field of a Dedekind domain $R$, and $S$ is its ...
2
votes
4answers
59 views

Field extension trace/norm confusion pertaining to multiplication matrix

I've gotten stuck on page 37 on P Samuel's 'Algebraic Theory of Numbers', on an equation that also features at the start of chapter 12 of Ireland and Rosen. The setup is, if we have a field $K$, and ...
1
vote
0answers
27 views

$M\cong N$ iff $[M:N]_R$ is a principal fractional ideal

Let $R$ be a Dedekind ring, $K$ its field of fractions, $U$ a finite vector space over $K$, and $M,N$ finitely generated $R$-modules that span $U$, i.e. contain a basis of $U$. For every $\mathfrak p ...
1
vote
1answer
73 views

Questions about a proof in Greenberg's Book.

I am trying to understand the proof of the following lemma : Lemma ' : Suppose that $X$ is a finitely generated $\Lambda$-module ($\Lambda =\mathbb Z_p[[T]]$) and that ...
0
votes
2answers
30 views

Meaning of open (and closed) subgroup

In Cassels and Froehlich's ANT, I met with the following statement: p. 5: The additive group of $K$ is the union of open (and hence closed) subgroups $\mathfrak p^n$ ($n\in \Bbb Z$), whose ...
1
vote
2answers
59 views

Applications of additive version of Hilbert's theorem 90

Additive version of Hilbert's theorem 90 says that whenever $k \subset F$ is cyclic Galois extension with Galois group generated by $g$, and $a$ is element of $L$ with trace 0, there exists an ...
0
votes
2answers
92 views

$\sqrt{10}$ is an irreducible element in the integral domain $\mathbb{Z}+\mathbb{Z}\sqrt{10}$

How can we prove that $\sqrt{10}$ is an irreducible element in the integral domain $\mathbb{Z}+\mathbb{Z}\sqrt{10}$? I think in general this is true for $\sqrt{m}$ where $m$ is not a perfect square ...
2
votes
1answer
96 views

Units in quadratic field.

Let $p$ be a prime number, such that $p\equiv 1\pmod 8.$ and $K=\mathbb Q(\sqrt p).$ Can Someone help me to prove this : $$\epsilon=a+b\sqrt p\hspace{4mm}a,b\in \mathbb Z\hspace{2mm}\text{is a ...