Questions related to the algebraic structure of algebraic integers

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2
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1answer
62 views

Fermat's Last Theorem and the Projective Curve $C_N$

In "Silverman & Tate" on page 230 in the appendix on projective geometry, there is the remark: The $N$th Fermat curve $C_N$ is the projective curve: $$C_N: X^N + Y^N = Z^N$$ ...
3
votes
1answer
38 views

Two question about ${\mathbb Q}(\alpha)$ for $\alpha$ := $\sqrt[n]{7}$ + $\sqrt[n+3]{7}$

Let $\alpha$ = $\sqrt[n]{7}$ + $\sqrt[n+3]{7}$ with $n$ not divisible by $3$. Prove that $[{\mathbb Q}(\alpha) : {\mathbb Q}] = n(n + 3)$. Conclude that $\alpha$ is constructible if and only if $n = ...
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2answers
33 views

Is there a subset of natural numbers with a special property

Let set $A$ be an infinite big subset of the set $\mathbb{N}$ (set of natural numbers),it is not equal to $\mathbb{N}$ and it has the following property: For every $a$ that is not from the set $A$ ...
0
votes
1answer
150 views

Prove generalisation of the Tower Law

I need to prove the following generalisation of the Tower Law: Let $L/K$ be an extension of fields, and $V$ a non-zero vector space over $L$. Then $V$ is finite-dimensional over $K$ if and only if ...
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0answers
36 views

Do you know any answer for equation y^2 = x^3 + k? [duplicate]

As you know, the equation y^2 = x^3 + k for k like (4n-1)^3 - 4m^2 that m , n are integers & no prime number that p is congruent to 1 modulo 4 count m, don't have any answer & it's proof is by ...
6
votes
4answers
107 views

Uniqueness of representation of prime as $x^2+2y^2$

It can be proven that every prime $p\equiv1,3\mod{8}$ can be written in the form $a^2+2b^2$. Is it true that this representation is unique? This is certainly true for primes written in the form ...
5
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3answers
72 views

Eisenstein integers and applications to Diophantine equations

Solve the equation $7\times 13\times 19=a^2-ab+b^2$ for integers $a>b>0$. How many are there such solutions $(a,b)$? I know that $a^2-ab+b^2$ is the norm of the Eisentein integer $z=a+b\omega$, ...
0
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1answer
41 views

Taylor expansion in $p$-adic integers

Let $f \in Z_p[X]$, then for $ x, y \in Z_p$, $\exists a \in Z_p$ s.t. $f(y)=f(x)+(y-x)f'(x)+(y-x)^2a$. Why is Taylor formula applicable to polynomial in $p$-adic integers $Z_p$? What condition ...
2
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0answers
62 views

Prove that for $n\ge 6$ there is always a solution

We have the eation $\frac{1}{a_1^2} + \frac{1}{a_2^2}+...+\frac{1}{a_n^2}=1$. Prove that the equation has for $n\ge 6$ always natural solutions. Any $\frac{1}{x^2}$ can be displayed as sum of 4 ...
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2answers
118 views

Topology on $Z_p$

let $Z_p$ denote the $p$-adic integers, then it has a topology as a subspace of $\prod_nZ/p^nZ$, where $Z/p^nZ$ is given the discrete topology. (reference I posted before: Why $Z_p$ is closed.) Now ...
0
votes
1answer
46 views

Constructing idele from a rational number.

I am a novice to concept of idele ,despite the fact that I have gone through all its expositions in standard literature. Excusing my ignorance,suppose I take q=396000. Does it mean that the idele q= ...
2
votes
2answers
39 views

Group homology with coefficients vanish

Say $G$ is a group and $M$ is a $\mathbb ZG$-module with the property that $H_i(G;M)=0$ for all $i\ge 0$. Does this happen besides $M=0$?
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2answers
39 views

Solve $\frac{x}{m}=\lfloor{x^{2/3}}\rfloor+ \lfloor{x^{1/2}}\rfloor+1$

Prove that for every natural number $m$ there is a natural solution for the eqation $\frac{x}{m}=\lfloor{x^{2/3}}\rfloor+ \lfloor{x^{1/2}}\rfloor+1$ Beside the typical inequality I can't get nothing ...
1
vote
0answers
40 views

Leads for penetrating the field of Algebraic number theory

I need to rapidly get up to speed on the following topics, for the purposes of an internship: Global and local fields. Localization. Number fields, function fields, etc. Ring of integers, field of ...
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2answers
69 views

Why $Z_p$ is closed.

Let $A_n=\mathbb{Z}/p^n\mathbb{Z}$ be a ring and $p$ is prime, $\phi_n: A_n\rightarrow A_{n-1}$ be a natural homomorphism (Elements of $A_{n}$ define in an obvious way elements of $A_{n-1}$). Define ...
3
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1answer
53 views

Special case of Kronecker–Weber theorem.

Let $K$ be a number field contained in $m^{th}$ cyclotomic field, that is $K \subset \Bbb{Q}(\omega)$ where $\omega$ is a primitive $m^{th}$ root of unity. Let $p^k$ be the exact power of a prime $p$ ...
11
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3answers
95 views

Prove that $2$, $3$, $1+ \sqrt{-5}$, and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$.

So the Norm for an element $\alpha = a + b\sqrt{-5}$ in $\mathbb{Z}[\sqrt{-5}]$ is defined as $N(\alpha) = a^2 + 5b^2$ and so i argue by contradiction assume there exists $\alpha$ such that $N(\alpha) ...
4
votes
2answers
83 views

A question about rings of algebraic integers

Let $R$ be a subring of the field of algebraic numbers. If $R\cap \mathbb{Q}= \mathbb{Z}$, does it follow that all of the elements of $R$ are algebraic integers?
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0answers
43 views

Exercise 1.10 from Silverman “The Arithmetic of Elliptic Curves ”

I am having trouble with Silverman's exercise 1.10(b). The converse of (a) is easy because there is no integer solution to the equation when $p \equiv 3$ mod $4$. However, this method does not work ...
4
votes
3answers
190 views

Find the number of integral solutions of $(x,y)$

Given this equation: $4x^3+5=y^2$ Find the ordered pairs of $(x,y)$ where $x,y\in Z$
0
votes
1answer
27 views

The intersection of a and b is a superset of the product when a and b are ideals

Let a and b be ideals of a ring A. Define $$ab=\left\{{\sum_{j=1}^{n} a_jb_j|a_j\in a,b_j \in b,n \in \mathbb{N}}\right\}$$ Prove that $ab$ and $a\cap b$ are ideals of A, and that $a\cap b \supseteq ...
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0answers
19 views

Ideals of the quotient ring of A [duplicate]

Let A be a ring and b be an ideal of A. The quotient ring of A by b, denoted A/b is the ring of all equivalence classes A + b. Prove that the assignment $$c → c/b$$ induces a one-to-one ...
2
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0answers
26 views

Definiton of topology in Galois group

In the book Algebraic numbers and algebraic functions of E. Artin, chapter six, page 103-104. Artin never says that $\Omega$ is of characteristic $0$ or $p>0$. 1) But by defining of ...
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votes
2answers
101 views

Conceptual reason why a quadratic field has $-1$ as a norm if and only if it is a subfield of a $\mathbb{Z}/4$ extension?

I have convinced myself in a computation-heavy, ad-hoc way that a quadratic extension $K$ of $\mathbb{Q}$ occurs as the unique quadratic subfield of a $\mathbb{Z}/4$-extension of $\mathbb{Q}$ if and ...
1
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1answer
324 views

Discriminant of a binary quadratic form and an order of a quadratic number field

Let $ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. Let $D = b^2 - 4ac$ be its discriminant. It is easy to see that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). Conversely ...
2
votes
1answer
96 views

Solutions to $y^2 = x^3 + k$?

As you know, the equation $y^2 = x^3 + k$ for $k = (4n-1)^3 - 4m^2$, with $m, n \in \mathbb{N}$ and no prime number that p is congruent to 1 modulo 4 count m, don't have any answer and its proof can ...
1
vote
1answer
27 views

algebraic conjugate and sum of roots of unity

In above lemma, why $|a'| \leq 1$ still holds? I didn't see how it relates to "algebraic conjugate of a root of unity is also a root of unity", since $a$ is the sum of unity. (definition of ...
9
votes
2answers
229 views

What are some equivalent statements of (strong) Goldbach Conjecture?

What are some equivalent statements of (strong) Goldbach Conjecture ? We all know that Riemann Hypothesis has some interesting equivalent statements. My favorites are involved with Mertens ...
1
vote
1answer
59 views

Fundamental Unit In Algebraic Fields

Say we have an algebraic field with an infinite amount of units. If one multiplies two units one obtains another unit. In some cases, all units are powers of just one unit ( that's the fundamental ...
11
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1answer
150 views

Is $\sum\frac1{p^{1+ 1/p}}$ divergent?

Is $\displaystyle\sum\frac1{p^{1+ 1/p}}$ divergent? How can we prove that it is divergent or convergent in analytic number theory? I know what bound of the n-th prime number is, and that its order is ...
15
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2answers
411 views

What would change in mathematics if we knew $\pi+e$ is rational?

It is well known that there's no conclusion now whether $\pi+e$ is rational or not. What would happen if we knew that $\pi+e$ is rational? Specifically, are there related open problems that would be ...
1
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1answer
22 views

Are the discriminant of abelian cubic extensions of $\Bbb Q$ equal to the square of their conductor?

Here the conductor $N$ of an abelian extension $\Bbb Q \subset K$ is the smallest positive integer $N$ such that $K \subset \Bbb Q(\zeta_N)$. Thanks to class field theory there is an equivalence ...
1
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1answer
58 views

$\sqrt {-6}$ is not prime in $\mathbb{Z}+\mathbb{Z}\sqrt {-6}$

Suppose $\sqrt{-6}|(a+b\sqrt{-6})(c+d\sqrt{-6})$. I need to show that $\sqrt{-6}$ does not divide $(a+b\sqrt{-6})$ and does not divide $(c+d\sqrt{-6})$. I thought you might arrive at some ...
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1answer
33 views

Basis for rank $n$ ring containing $1$.

Suppose $L$ is a finite (separable?) extension of a number field $K$, $\mathcal{O}_K$ is the ring of integers of $K$, and $\mathcal{O}_L$ is the integral closure of $\mathcal{O}_K$ in $L$. How can ...
2
votes
1answer
81 views

Algebraic number theory exercise

I'm trying to do the following exercise from Neukirch's Algebraic Number Theory (exercise 2, $\S 1$, chapter 1): Show that, in the ring $\mathbb{Z}[i]$, the relation ...
15
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2answers
253 views

So what *is* the Euclidean function for $\mathcal{O}_{\mathbb{Q}(\sqrt{69})}$?

It's my understanding that $\mathcal{O}_{\mathbb{Q}(\sqrt{69})}$ is UFD, PID, and Euclidean, but not norm-Euclidean. If it were norm-Euclidean, there would be a solution to $28 = q \left(\frac{5}{2} ...
46
votes
5answers
4k views

Are all algebraic integers with absolute value 1 roots of unity?

If we have an algebraic number α with (complex) absolute value 1, it does not follow that α is a root of unity (i.e., that αn=1 for some n). For example, (3/5 + 4/5 i) is not a root of ...
2
votes
1answer
59 views

Prove Composite Infinitely Often

Suppose we let $f(x) = ax^2 + bx + c$ be a non-constant polynomial, and assume that $a$, $b$, and $c$ are integers. Prove: There are infinitely many integers $n$ such that $f(n)$ is composite.
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1answer
41 views

Classification of Discrete Subrings of $\mathbb C$

I am interesting in classifying the subrings of $\mathbb C$ which are discrete with respect to the standard topology (that is, the topology induced by the standard absolute value). Here, I am using ...
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2answers
536 views
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1answer
31 views

Cover of $\Bbb P^1_k$ ($k$ sep. closed) unramified away from $\infty$ and tamely ramified at $\infty$

I'm reading a paper where the authors claim that for a separably closed field $k$ of characteristic $p>0$, there's no cover of $\Bbb P^1_k$ unramified away from $\infty$ and tamely ramified over ...
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votes
0answers
48 views

Integration over ideles over $\Bbb{Q}$ , Tate`s thesis special case

Let $f\in S(A_\Bbb{Q})$ that is $f$ is adelic Schwartz-Bruhat function over $\Bbb{Q}$, such that all its components in the finite places are characteristic functions of the corresponding ring of ...
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0answers
46 views

Tate thesis : Global functional equation [closed]

It will be very helpful if someone tells me how to do EXERCISE 1 here. I have done part $2$. I cannot do part $1$ and part $3$. I tried part $1$ by decomposing $Z(f,s)$ as the product of its local ...
3
votes
1answer
36 views

Explicitly computing uniformisers of local fields

Consider the field tower $L/K'/K$ where $L=\mathbb{Q_3}(\xi,2^{1/3})$, $K'=\mathbb{Q_3}(\xi) $ and $K=\mathbb{Q_3}$. Here, $\xi$ is a primitive cube root of unity, and $\mathbb{Q_3}$ is the 3-adics. ...
3
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0answers
59 views

Any good supplementary text on Algebraic Number Theory?

I'm going to study algebraic number theory by myself with the following texts: Algebraic Number Theory by Cassels, Frolich Number Fields by Daniel Marcus But I'm not sure whether or not I should ...
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1answer
37 views

What is a system of representatives of the residue field in its ring R?

Let R be a complete discrete valuation ring, with field of fractions K and residue field $\hat{K}$. Let S be a system of representatives of $\hat{K}$ in R. Can someone please explain to me what a ...
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0answers
19 views

Prolonging a discrete valuation in Serre's Local Fields?

I am really struggling with the concept of prolonging a valuation. Can someone please explain what 'e(E'/K)' is in the exercise below, what it means for K to be complete under a discrete valuation and ...
1
vote
1answer
26 views

Show that the idele group of a number field is locally compact

Let $k$ be a number field and $M_k$ the canonical set of places of $k$. Also let $S_\infty$ be the set of Archmedean places of $k$. For each $v\in M_K$ let $k_v$ be the completion of $k$ wrt an ...
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0answers
17 views

Reference request for valuations in algebraic number theory

I am going through the book "Primes of the form $x^2$ + n$y^2$" and I understand the background material about algebraic number fields and class numbers(from Ireland and Rosen). However, I do not have ...
2
votes
1answer
41 views

How many Fermat tests are needed to verify a Carmichael number

If $n$ is a Carmichael number, then for all values $a$ such that $0<a<n$ (and $a \perp n$): $a^{n-1} \equiv 1 \mod n$ However, is it not necessary to check check all $a$ values because for a ...