Tagged Questions

Questions related to the algebraic structure of algebraic integers

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1
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1answer
22 views

Prove that if $f\in R[X]$ , then $\displaystyle\prod _{\sigma \in G}f^{\sigma}\in \mathbb{Z}[X].$

Let $K$ be an algebraic number field and $R$ be the ring of algebraic integers of $K.$ Denote by $h^{\sigma}$ the polynomial obtained from $h\in K[X]$ after applying to its coefficients the ...
5
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1answer
52 views

likely open number theory problem: finite sum of $\zeta(2)$ equal to a square of rationals

Which $n$ can let $S=1+\frac14+\frac19+\cdots+\frac1{n^2}$ be a square of a rational number? Obviously, $1$ and $3$ work, but how to prove they are the only ones? I think this problem is really hard. ...
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0answers
5 views

$R_S (=K \cap A_{K,S})$ is a Dedekind domain

Let $K$ be a global field and let $S$ be a finite, nonempty set of places of $K$ containing the infinite ones. Show that $R_S (=K \cap A_{K,S})$, the ring of $ S-$ integers of $K$, is a Dedekind ...
2
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1answer
36 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 ...
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0answers
11 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 ...
-1
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0answers
41 views

Existence of dual basis in Algebraic Number Theory [on hold]

Suppose we have a non-degenerate symmetric bilinear form on an n-dimensional vector space $V$, written for convenience using inner product notation $(x, y)$. If $x_1, \ldots, x_n$ is any basis for ...
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0answers
9 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 $K$.
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0answers
28 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 ...
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0answers
26 views

Why does Vinogradov method fail to prove even number Goldbach Conjecture? [on hold]

Why does Vinogradov method to prove three prime theory fail to prove even number Goldbach Conjecture ? Will multi-variable Trigonometric Sums be able overcome the difficulties ?
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0answers
25 views

what are some attempts to modify Vinogradov's methods to prove Three Prime Theory to Even Number Goldbach Conjecture?

All: What are some attempts to modify Vinogradov's methods to prove Three Prime Theory to Even Number Goldbach Conjecture ? I assume that there must be many such attempts, I tried to understand ...
0
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0answers
10 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
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2answers
39 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 = ...
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0answers
50 views
+50

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
14 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 ...
5
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4answers
891 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} $. ...
1
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1answer
27 views

Solution of Pell equation over field of p-adic numbers

Right now I am studying Pell equation. Using continued fractions, we can find the solution of Pell equation. Now my question, is it possible for me to find a solution in the field of p-adic numbers ...
5
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1answer
66 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 ...
2
votes
3answers
70 views

Solve in integers: $x^2 = y^2 + y + 1$

Solve this equation in integers: $$x^2 = y^2 + y + 1$$ I know $2$ ways to solve this. But they are not easy. Maybe there is some quick method.
1
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2answers
89 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
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1answer
32 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
13 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
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1answer
28 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
54 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
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1answer
39 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
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1answer
24 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 $\mathbb{Z}[\zeta]$, $p$ ...
1
vote
4answers
40 views

$\mathfrak{a}_{1} + \dots + \mathfrak{a}_{n} = A \Rightarrow \mathfrak{a}_{1}^{r_{1}} + \dots + \mathfrak{a}_{n}^{r_{n}} = A$

I have to prove the following : Let $A$ be a commutative ring with unity and let $\mathfrak{a}_{i}$ be ideals in $A$. Assume that $\mathfrak{a}_{1} + \dots + \mathfrak{a}_{n} = A$. Let $r_{i}$ be ...
2
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0answers
30 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. ...
2
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1answer
47 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
38 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
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1answer
127 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
44 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
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1answer
58 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 ...
5
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0answers
85 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
51 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
40 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 ...
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votes
1answer
150 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
47 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
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1answer
21 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
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0answers
67 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 ...
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0answers
72 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 ...
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0answers
53 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
40 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
49 views
6
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1answer
70 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: ...
1
vote
1answer
38 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
146 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
64 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
64 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 $\mathbb{Z}[\zeta]^{\times} = \left<\zeta\right>\mathbb{Z}[\zeta + \zeta^{-1}]^{\times}$. ...
4
votes
1answer
64 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 ...
1
vote
1answer
29 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 ...