Questions related to the algebraic structure of algebraic integers

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2
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1answer
58 views

Periodicity over the prime indices of a multiplicative sequence implies periodicity?

I have a real sequence $(a_p)$ indexed by the prime numbers which takes values -1, 0, or 1, having the property that $a_p=a_q$ whenever $p\equiv q \pmod m$, where $m$ is a fixed integer $>2$. I'm ...
1
vote
0answers
18 views

Show that a Field Extension is Unramified Using Transitivity

Let $K=\mathbb{Q(\sqrt{5})}, L=\mathbb{Q(\sqrt{7})}, M=\mathbb{Q(\sqrt{35})}$, and $KL=\mathbb{Q(\sqrt{5},\sqrt{7})}$. Show that $KL/M$ is unramified (i.e. every prime ideal of $M$ is unramified in ...
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0answers
13 views

Root of unity in a global field

Let $K$ be a global field. How to show that $|x|_v= 1$ at every place $v$ of $K$ if and only if $x$ is a root of unity in $K$.
5
votes
1answer
198 views

$\Bbb{R}/n\Bbb{Z}$ is isomorphic to $A_\Bbb{Q}/(\Bbb{Q}+C_n)$.

Let $A_\Bbb{Q}$ be the adele group of $\Bbb{Q}$. Let $C_n=\{x \in A_\Bbb{Q}: x_\infty=0 \text{ and }x_p \in p^{\operatorname{ord}_p(n)}\Bbb{Z}_p \text{ for prime }p\}$. I want to show that ...
0
votes
0answers
25 views

Norm in cyclotomic field

Suppose $p$ is a rational prime and $\zeta=e^{2\pi i/ p}$. Prove that the groupp of non-zero elements of $\mathbb Z_p$ is cyclic, show that there exists a monomorphism $\sigma:\mathbb Q(\zeta)\to ...
2
votes
2answers
60 views

Splitting of Primes in a Given Field

Find how $p=2,3,5,7$ splits in $\mathbb{Q}(\sqrt{-5})$ (i.e. find those $e_i,f_i$ for $1 \leq i \leq r$). Can somebody please explain how this is done? My attempt is the following: Let K = ...
0
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0answers
107 views

Rational points of $ax^2+by^2=z^r$, $r $ odd integer.

I am trying to find the rational points of:$$ax^2+by^2=z^r$$ I am aware that:$$(u^r-2^{r-2}v^r)^2+(2uv)^r=(u^r+2^{r-2}v^r)^2$$ How can I deduct the results?
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0answers
26 views

Questions on Heath-Brown's paper “Kummer’s Conjecture for Cubic Gauss Sums”

On page 21 in Heath-Brown's paper "Kummer’s Conjecture for Cubic Gauss Sums" (http://eprints.maths.ox.ac.uk/158/1/kummer.pdf), a formula says $$\sum_{j\in \mathbb{Z}[\omega]}f(j)=\sum_{k\in ...
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4answers
953 views

How to factorise a number in $\mathbb {Z}[\sqrt {-5}]$?

I am studying quadratic number fields. I have a question about factorization in $\mathbb {Z}[\sqrt {-5}]$ which seems less trivial than factorization in the Gaussian integers. Let $ w=\sqrt {-5} $. ...
5
votes
1answer
79 views

Showing two elements are coprime in a ring of integers

Let $\alpha$ and $\beta$ be the two roots of the polynomial $x^2 - x + 2$. I was wondering if someone could explain to me why $(y - \alpha)$ and $(y - \beta)$ are coprime (for any integer $y$) in the ...
1
vote
2answers
107 views

Torsion free module over a PID is flat

Suppose a ring of integers $S$ is an extension of a ring of integers $R$ with $\mathfrak{q}$ a prime ideal in $S$ and $\mathfrak{p}=\mathfrak{q}^c$ in $R$. Is there a straightforward way of showing ...
0
votes
1answer
43 views

Is an extension of a discrete absolute value discrete too?

Suppose $L/K$ is a finite extension of fields, suppose $v$ is a non-archimedean absolute value on $L$ such that the restriction of $v$ on $K$ is non-trivial and discrete. Can we say that $v$ is ...
0
votes
1answer
18 views

Gaussian sums values

I have the following problem: Denoting $S(q,a,\chi ) = \sum_{x=1}^q \chi (x) e(ax/q)$, where $\chi $ is an arbitrary character modulo $q$, I have to prove $$\sum_{a=1}^q \vert S(q,a,\chi ) \vert ^2 = ...
2
votes
1answer
30 views

Basis for sum of all extensions of a completion: $\bigoplus_{w \mid v} \mathcal{O}_{w}$ over $\mathcal{O}_{v}$

I was going over notes from a class and it was stated (without proof) that if $\xi_{1}, \ldots, \xi_{n}$ is a basis of $K/k$, then for almost all places $v$, $\xi_{1}, \ldots, \xi_{n}$ is a basis for ...
2
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0answers
67 views

Mellin transform on $\mathbb{Z}[\omega]$

Let $\omega=\frac{-1+i\sqrt{3}}{2}$ be a complex cube root of unity. The Eisenstein integers $\mathbb{Z}[\omega]$ (a unique factorization domain) are of the forms $a+b\omega$ where $a$ and $b$ are ...
3
votes
1answer
49 views

Calculating zeta functions over a field

I am learning about zeta functions and have been trying the following example: Calculate the zata function of $x_0x_1-x_2x_3$ over $\mathbb{F}_p$. Does there exist an easy formula for calculating ...
4
votes
1answer
41 views

Inertia Degree in Cyclotomic Extensions

Let $\zeta$ be a primitive $l$th root of unity, where $l$ is prime. If $p$ is another prime number, let $f$ be the order of $p$ in $U(\mathbb{Z}/l \mathbb{Z})$. Then in $\mathbb{Z}[\zeta]$, $p$ ...
2
votes
0answers
35 views

Find the ring of algebraic integers. [duplicate]

Find the ring of algebraic integers in $K=\mathbb Q(\sqrt[3]{2})$. So, I know that $K=\{a+b\sqrt[3]{2}+c\sqrt[3]{2}^2 \mid a,b,c \in \mathbb Q\}$. My professor has done very little on this topic. ...
3
votes
2answers
105 views

$\overline{\mathbb{Z}}$ is not a Dedekind domain.

I have to prove the following statement: Let $\overline{\mathbb{Z}}$ be the ring of all algebraic integers in (a fixed choice of) $\overline{\mathbb{Q}}$. Then $\overline{\mathbb{Z}}$ is not a ...
6
votes
2answers
45 views

Question about Thm 1.4.4 in ANT Alaca/Williams

I am studying Introductory ANT by Alaca/Williams, p12, theorem 1.4.4: "Let $m$ be a nonsquare integer such that $\mathbb {Z}+\mathbb{Z}\sqrt {m}$ is a PID. Let $p $ be an odd prime for which the ...
2
votes
1answer
129 views

Some Galois theory

I have a question on field extensions, and I can't seem to find precise answers when browsing through online notes etc. Here it is: suppose $K$ and $k$ are fields with $k \leq K$ and $[K : k] = m$ ...
2
votes
1answer
51 views

Non-Galois number fields and complex embeddings

Let $K$ be a number field. $K$ is a normal extension of $\mathbb{Q}$ iff $\exists f(x)\in\mathbb{Q}[x]: K$ is the splitting field for $f(x)$. A field extension is Galois iff it is normal and ...
2
votes
1answer
64 views

Finding the Discriminant of $f(x)=x^n+ax+b$ Using Differentiation

Greetings fellow Mathematics enthusiasts. I was hoping someone could offer me some advice on proving the following statement about the discriminant of a polynomial with degree $n$. Let ...
6
votes
0answers
123 views

Generalized class group of $\mathbb Q(\sqrt{-5})$

I follow the notation of Georges Gras: Class Field Theory, some of which I recall for convenience; feel free to skip the following lines if you are familiar with the notation. Let $K$ be a number ...
1
vote
1answer
55 views

Find all positive integer pairs $(x,y)$ and $(u,v)$ with certain relations.

Is there exists any positive integer pairs $(x,y)$ and $(u,v)$ for which, the relations, $x^2+y^2=u^2+v^2$ and $x^3+y^3=u^3+v^3$ are satisfied simultaneously?
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0answers
42 views

$(X+4)=X^4+X^3+X^2+X+1 \pmod{5}$ by ramification of prime ideals

In Milne's algebraic number theory notes, on page 65, there is the following example: $X^4+X^3+X^2+X+1\equiv(X+4)^4\pmod{5}$. And Milne asks: Why is that obvious? This comes after discussion of ...
7
votes
1answer
165 views

How does (21) factor into prime ideals in the ring $\mathbb{Z}[\sqrt{-5}]$?

The text of the exercise is the following: Show that $\mathbb{Z}[\sqrt{-5}]$ is a Dedekind domain, and that the identities $21 = (4+\sqrt{−5}) \cdot (4 − \sqrt{−5})$ and $21 = 3 · 7$ represent two ...
0
votes
2answers
53 views

Prove all elements of $A$ is algebraic over $C$, if all elements of $A$ are algebraic over $B$ and $B$ are algebraic over $C$

Let there be 3 fields $A$, $B$ and $C$. If all elements of $A$ are algebraic over $B$ and all elements of $B$ are algebraic over $C$, prove that this implies that all elements of $A$ is algebraic ...
1
vote
1answer
36 views

Prime containing ideal in number ring divides the index of ideal

I'm working through Peter Stevenhagen's notes on Algebraic Number Theory, and the third section starts: In order to factor an ideal $I$ in a number ring $R$ [into its primary composition $I = ...
6
votes
0answers
89 views

Name of a certain set

I want to know if there is any already-standard way to refer to the sets described as follows. For a set $X$, let $-X = \{-x: x \in X \}$; call it the negative of $X$. Take the set of all primes in ...
1
vote
0answers
89 views

What are the “hidden” symmetries in Goldbach Conjecture?

What are the "hidden" symmetries in Goldbach Conjecture ? If Goldback conjecture is true, the basic instinct is that there must exist some "symmetries" which ensure (and lead) such properties. As we ...
2
votes
0answers
76 views

What are the missing gaps to prove Goldbach Conjecture?

When Andrew Wiles proved FLT, all he needed to do was to prove "semi-stable elliptic curve case" of Shimura-Taniyama conjecture. He did not need to start from scratch, he just needed to fill this ...
2
votes
1answer
65 views

Ideals, Dedekind domain and $\mathbb{Z}[\sqrt{-3}]$

I have the ideal $\mathfrak{a} = (2, 1 + \sqrt{-3})$ in $\mathbb{Z}[\sqrt{-3}]$. I have to show that $\mathfrak{a} \neq (2)$ but $\mathfrak{a}^{2} = (2)\mathfrak{a}$ and then conclude that ideals do ...
1
vote
2answers
66 views
7
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1answer
79 views

Generalizing the Big Omega function to Integral Domains

The $\Omega(n)$ function counts the total number of prime factors of $n$ counting multiplicity. Obviously, this definition extends to any Unique Factorization Domain. I have two follow up questions: ...
2
votes
1answer
45 views

Showing $\mathbb{Z}+\mathbb{Z}(\frac{1+\sqrt m}{2})$ is a Euclidean domain

Does anyone know an elementary proof for the following proposition? It is stated without proof in my textbook: Let $m$ be a negative squarefree ineteger with $m = 1 \pmod 4$. Then the integral domain ...
5
votes
1answer
153 views

Linear independence of fractional powers

Reflecting on this recent MSE question, I was led to the following conjecture : Let $A=\left\lbrace x^y \mid x,y\in{\mathbb Q}_+ \right\rbrace$. If $\alpha,\beta,\gamma\in A$ are pairwise $\mathbb ...
2
votes
1answer
68 views

Largest ideal of a local field on which a character is trivial

Let $K$ be a nondiscrete locally compact field. Then fixing a character $\chi$ on $K$, any character on $K$ can be written as $t \mapsto \chi(xt)$ for some $x \in K$. For $E \leq K$ a closed ...
3
votes
1answer
76 views

Group of Units in Cyclotomic Integers

I'm trying to show that for any $p$-th root of unity $\zeta$, where $p$ is an odd prime, we have $\mathbb{Z}[\zeta]^{\times} = \left<\zeta\right>\mathbb{Z}[\zeta + \zeta^{-1}]^{\times}$. ...
7
votes
1answer
114 views

Computing a uniformizer in a totally ramified extension of $\mathbb{Q}_p$.

Do you know how to compute a uniformizer of $\mathbb{Q}_p(\zeta_{p^n},p^\frac{1}{p})$? Where $\zeta_{p^n}$ is a primitive $p^n$-th root of 1 and $p$ is an odd prime.
4
votes
1answer
77 views

What was Lame's proof?

In 1847, Lame gave a false proof of Fermat's Last Theorem by assuming that $\mathbb{Z}[r]$ is a UFD where $r$ is a primitive $p$th root of unity. The best description I've found is in the book ...
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vote
1answer
43 views

$\mathbb{Q}(\sqrt{m}, \sqrt{n})$ : ring of integers, integral basis and discriminant

In the following document, http://people.math.carleton.ca/~williams/papers/pdf/033.pdf, I found three results about biquadratic fields and their ring of integers. It's the proof of the first theorem ...
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vote
2answers
41 views

Prove that $\mathbb{Z}[\zeta_{p} + \zeta_{p}^{-1}]$ is the ring of integers of $\mathbb{Q}(\zeta_{p} + \zeta_{p}^{-1})$

I'm a bit at a loss about what I can say in this situation. Do I have to show that $\zeta_{p} + \zeta_{p}^{-1}$ form an integral basis ? If I do, I have no idea how to do it. If not, can I use the ...
4
votes
3answers
90 views

How to multiply and reduce ideals in quadratic number ring.

I am studying quadratic number rings and I have a problem with multiplying and reducing ideals, for example: Let $w=\sqrt{-14}$. Let $a=(5+w,2+w)$, $b=(4+w,2-w)$ be ideals in $\mathbb Z[w]$. Now, ...
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votes
0answers
25 views

Norm and trace in a number field

In the book's Algebraic number theroy Ian Stewart, exercise 12 chapter two. $K$ is a number field, $N_K,T_K$ is a norma and trace. 12: Give examples to show that for fixed $\alpha$, $N_K(\alpha)$ ...
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1answer
35 views

Algebraic Integers and Irreducible Polynomials

Let $\alpha$ be an algebraic integer and let $f$ be a monic polynomial over $\mathbb{Z}$ of least degree having $\alpha$ as a root. Prove that $f$ is irreducible. I am having so many troubles with ...
3
votes
0answers
29 views

The irreducibility of the polynomial $x^p\pm px-t$, where $t$ is an integer, $p$ is a prime number.

Let $f(x)=x^p+px+1$, where $p$ is an odd prime. Prove that $f(x)$ is irreducible. This is an exercise of a course (linear algebra, the first chapter focus on the polynomial rings), and the exercise ...
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votes
0answers
26 views

Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ , $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$, what is the intersection?

Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ and $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$, What is the intersection $F_\infty\cap K_\infty$? Here $\zeta_{2^n}$ is a ...
2
votes
0answers
54 views

Algebraic Integers

If $\alpha$ and $\beta$ are algebraic integers then show $\alpha + \beta$ and $\alpha \times \beta$ are both algebraic integers. I know that an algebraic integer is a root of some monic polynomial ...
2
votes
1answer
70 views

Are all transcendental numbers theoretically accessible?

I apologize if the title (and the body) of this question is worded incorrectly, but I have no real experience in (transcendental) number theory, so it's probably the best I can do. I've been thinking ...