1
vote
0answers
32 views

Shtukas?$\mbox{}$

Does there exist an exposition of the significance of shtukas for someone who is mathematically literate but is largelly ignorant of Drinfeld modules? This arises in the work of Peter Scholze among ...
4
votes
2answers
52 views

what is the most easy to read Algebraic Geometry book? [duplicate]

All: what is the most easy to read (most accessible) Algebraic Geometry book ? (If possible, I am looking for an introduction book, maybe for undergraduate, and maybe similar to A Friendly ...
4
votes
0answers
56 views

What are major algebraic number theory attempts, results and progressions toward Goldbach's Conjecture?

To my understanding, most progress toward Goldbach's Conjecture has been made in analytic number theory. Progress has often based on sieve, asymptotic estimation or other analytic methods. What are ...
7
votes
1answer
46 views

What is the smallest $d$ such that $4$ has more than one distinct factorization in $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$?

Or if there is no such $d$, how do I prove it? Obviously there is no point to looking for this in an UFD. I've looked in other rings, and each time I think I found it, I divide one of the factors by ...
2
votes
1answer
22 views

Problem with the hyperelliptic equation

Suppose $K$ is an algebraic number field with $ [ K : \mathbb{Q} ] = d $. $X, Y , \alpha_1 , \ldots \alpha_n $ are in $O_K$ , i.e. are integral over $\mathbb{Z} $. Suppose that we have the following ...
2
votes
0answers
52 views

Are there any fundamental new discoveries in Algebraic Number Theory than in ‘traditional’ number theory ? [closed]

I am new to Algebraic Number Theory. I wonder if there is any fundamental new discoveries in Algebraic Number Theory than in ‘traditional’ number theory ? I want to know, beside ‘generalizing’ or ...
0
votes
0answers
50 views

Which is the best book on Goldbach conjecture research

Is there a book which summarizes the major research results in the past, and current research trends, for the Goldbach conjecture? I know, much progress has been made in Analytic Number theory in ...
2
votes
0answers
144 views

$e=1$ in Theorem 30 from Marcus book “number fields”

Theorem 30 in Marcus book states that, if $p\in\mathbb Z$ is an odd prime and $q$ is a prime $\neq p$, then, fixing $d$ as a divisor of $p-1$ we have that $q$ is a $d$-th power $\operatorname{mod}q$ ...
0
votes
1answer
63 views

Can anyone recommend an easy to read algebraic number theory book?

Can anyone recommend an easy to read algebraic number theory book ? I prefer a book with good examples. (hints or answers to selected questions if possible. Not sure if it is possible for a book of ...
0
votes
1answer
44 views

The minimum number of digits after the floating-point, which uniquely identify every irrational square root

Let the following: $B:$ a natural number larger than $1$ $S:$ a set of irrational numbers in the range $(0,1)$ represented in base $B$ $L:$ the minimal prefix length which uniquely identifies every ...
1
vote
1answer
22 views

Bound on integral solutions to $ar^2+bs^2=m$

The problem is as follows. Let $m$ be a fixed integer. Let $a,b\geq0$ be integers such that $(a,b)=1$ and both $a$ and $b$ are square-free. I want to show that the set $\{r,s\in\mathbb ...
7
votes
1answer
51 views

An extension of an algebraic number field which makes an integral ideal $I$, a principal ideal

I want to show that, given an ideal $I \subseteq \mathcal O_K$ (where $K/\mathbb Q$ is an algebraic number field), there is a finite extension $K'/K$ such that, $I\mathcal O_{K'}$ becomes a principal ...
4
votes
3answers
88 views

Does there exist a cubic polynomial $f(x)$ such that $f(x)\equiv0 \pmod p $ has no integer solutions if $p\equiv 3\pmod 4$?

As we know that $f(x)=x^2+1\equiv0 \pmod p $ has no integer solutions if $p\equiv 3\pmod 4$, does there exist a cubic polynomial $f(x)=ax^3+bx^2+cx+d~(a,b,c,d \in\mathbb Z,a\neq 0) $ such that ...
0
votes
1answer
37 views

Dedekind rings which are UFDs but not PIDs?

I just have a really quick question of an example that I was trying to come up with. Are there any number rings which are UFDs but not PIDs?
2
votes
1answer
48 views

Algebraic Integers in $\mathbb{Q}(\sqrt{m})$ and Norms on them

I'm having a problem with a section of Niven's book the Theory Of Numbers. I am trying to show: If an integer $\alpha \in \mathbb{Q}(\sqrt{m})$ is neither zero nor a unit, prove that ...
3
votes
1answer
39 views

Non unique factorization domains with prime factorizations with differing number of primes

As is well-known, $Z[\sqrt{-5}]$ is not a ufd because $6$ has more than one prime factorization in this ring: $6=2\cdot 3$ and $6=(1+\sqrt{-5})(1-\sqrt{-5})$. But both of these prime factorizations ...
3
votes
0answers
48 views

Are there any primes that are never a factor of a Carmichael number?

Is there a prime number $p$ that $p > 2$, and in which $p$ is a never a factor of any Carmichael number $C_n$: (p ∤ $C_n$) Extended this to all numbers $m$, instead of just $p$, will prove the ...
28
votes
5answers
720 views

Can a Mersenne number ever be a Carmichael number?

Can a Mersenne number ever be a Carmichael number? More specifically, can a composite number $m$ of the form $2^n-1$ ever pass the test: $a^{m-1} \equiv 1 \mod m$ for all intergers $a >1$ ...
3
votes
2answers
54 views

Does it hold that the $p$-adic completion of the integers equals the completion of the localization in $p$?

maybe this is a stupid question, but I could not solve it even for the ordinary integers $\mathbb{Z}$. Furthermore, I don't have to much knowledge on algebraic number theory and ramifications. Let ...
3
votes
3answers
43 views

If algebraic $a$ has degree $n$, so does $-a$

I feel like the best way to move forward is to use a contradiction proof. Since $a$ is algebraic, and is of degree $n$, it has a minimal polynomial of degree $n$, so we can write ...
6
votes
2answers
62 views

minimal polynomial given an algebraic number

I am trying to find the minimal polynomial for the algebraic number $1+\sqrt{2}+\sqrt{3}$. My original thought was just let $\alpha=1+\sqrt{2}+\sqrt{3}$. The method I use though seems very ...
3
votes
2answers
57 views

explicit example of computing ray class field for imaginary quadratic?

Given an imaginary quadratic number field K, we can get its ray class field mod some ideal $\mathcal{m}$ by adjoining the j-invariant of an elliptic curve with complex multiplication given by ...
1
vote
2answers
52 views

Help with proof regarding degrees of polynomials

How do you prove that if $f(x)\mid g(x)$ in $F[x]$, then either $g(x) = 0$ or $\deg(g(x)) \geq \deg(f(x))$ I'm not really sure how to prove these types of statements
1
vote
1answer
48 views

When Is there no Local Power Integral Basis?

Let $A$ be a Dedekind domain, $K, L, B$ the usual designations of $A$'s quotient field, a finite separable extension of $K$, and integral closure of $A$ in $L$ respectively. If $\alpha \in B$ ...
45
votes
4answers
3k views

Fibonacci number that ends with 2014 zeros?

This problem is giving me the hardest time: Prove or disprove that there is a Fibonacci number that ends with 2014 zeros. I tried mathematical induction (for stronger statement that claims that ...
3
votes
1answer
51 views

Question on number theory ( related to (Z/p^rZ)* group )

This is a (different version to) question from Serre 'A Course in Arithmetic'.Let p be an odd prime number. $\forall n\geq 1$ (n positive integer), $f$ is defined by: $$f(n)=(-1)^n\prod_{1\le k\le n ...
1
vote
1answer
22 views

Extensions of nonarchimedean valuations

This is a question from Janusz 'Algebraic Number Theory'. Let $R$ be a DVR with maximal ideal $\mathfrak p=\pi R$. Let $K$ be the quotient field of $R$ and $\mid\cdot\mid_{\mathfrak p}$ the ...
3
votes
2answers
78 views

Finding the GCD of two Gaußian integers

How do you calculate the GCD of $6-17i$ and $18+i$ in $\Bbb Z [i]$?
3
votes
1answer
40 views

Showing two numbers are relatively prime in number fields

Solve $x^3-2=y^2$ in integers. The standard way to solve this problem is to consider the arithmetic of the ring of algebraic integers $\mathcal{O}_{\mathbb{Q}(\sqrt{-2})}$ and to show that ...
23
votes
1answer
250 views

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$?

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$? I think there are no such polynomials, but how to prove?
7
votes
1answer
123 views

Characterizing a sequence of primes

This is an attempt to finish up Characterizing the primes which don't divide any Pell-Lucas number(s) For primes $p \equiv 3 \pmod 4,$ there is always some solution to $x^2 - 2 y^2 = \pm 1$ with ...
5
votes
1answer
100 views

Units of $\mathbb{Z}[\sqrt[4]2]$

How would one compute the units in $\mathbb{Z}[\sqrt[4]2]$? According to one source, it can be shown that the fundamental units are $1 + \sqrt[4]2$ and $1 + \sqrt{2}$, but it does not specify the ...
2
votes
0answers
30 views

Meaning of tamely ramified extension.

Let $K$ be a complete field with respect to a discrete nonarchimedean valutaion. We denote $A$ and $\mathfrak{p}$ as its valuation ring and valuation ideal, respectively. For a finite Galois extension ...
3
votes
1answer
70 views

Determine Units of a Ring $\mathbb{Z}[\alpha]$

I am trying to determine the units of $Z[\alpha]$ where $\alpha$ satisfies the monic polynomial $\alpha^4+\alpha^3+\alpha^2+\alpha+1$. I found $Z[\alpha] := \lbrace a+b\alpha+c\alpha^2+d\alpha^3\;|\; ...
0
votes
1answer
55 views

Counting solutions mod p of a polynomial equation

Hello: Does somebody know if the following is true?: Let $f\in \mathbb{Z}[X]$ be a monic irreducible polynomial of degree $n$. Then there exists a positive integer $N$ and ...
1
vote
0answers
16 views

Logarithm of the basic Lubin-Tate formal group

Let $K$ be a local field with finite residue field of cardinality $q$. Let $\pi$ be a uniformizer. The basic Lubin-Tate group (associated to $\pi$) is the unique formal group associated to the ...
11
votes
5answers
1k views

How to tell if a Fibonacci number has an even or odd index

Given only $F_n$, that is the $n$th term of the Fibonacci sequence, how can you tell if $n \equiv 1 \mod 2$ or $n \equiv 0 \mod 2$? I know you can use the Pisano period, however if $n \equiv 1$ or ...
5
votes
4answers
149 views

When is the sum of two squares the sum of two cubes

When does $a^2+b^2 = c^3 +d^3$ for all integer values $(a, b, c, d) \ge 0$. I believe this only happens when: $a^2 = c^3 = e^6$ and $b^2 = d^3 = f^6$. With the following exception: $1^3+2^3 = 3^2 + ...
1
vote
0answers
50 views

Wolstenholme Number

Does Wolstenholme Numbers have perfect squares other than 1 and 49? The first few are 1, 5, 49, 205, 5269, 5369, 266681, 1077749 seems to be a complicated problem
1
vote
1answer
30 views

Valuation associated to a non-zero prime ideal of the ring of integers

I have a question from Frohlich & Taylor's book 'Algebraic Number Theory', p.64. I will keep the notation used there. Let $K$ be a number field, $\mathcal o$ its ring of integers. Let $\mathfrak ...
5
votes
0answers
80 views

27 lines on a smooth cubic surface

It is known that every smooth cubic surface with coefficients in $\mathbb{Q}$ has $27$ lines defined over a number field extension of $\mathbb{Q}$ of degree at most $51840$ as the group ...
5
votes
1answer
155 views

Serre's Modularity Conjecture — Weight

I was reading Serre's paper "Sur les Représentations Modulaires de Degré $2$ de Gal($\bar{\mathbb{Q}}/\mathbb{Q}$)" where he states his modularity conjecture (which is now a theorem). Following his ...
4
votes
2answers
99 views

The prime elements of the ring $ \mathbb{Z} \! \left[ \dfrac{i \sqrt{3} - 1}{2} \right] $.

I have to study the prime elements of the ring $ \mathbb{Z} \! \left[ \dfrac{i \sqrt{3} - 1}{2} \right] $. For the moment, I cannot find the general form of such elements. Can you help me? Thanks! :) ...
0
votes
1answer
41 views

Polynomial Diophantine Equations

So in general how does one decide if: $$ a_0 + a_1x + a_2x^2 ... a_{n_1}x^{n_1} = b_1y + b_2y^2 ... + b_{n_2}y^{n_k}$$ Has solutions for integers $x,y$ given real numbers $a_0, a_1. .. a_{n_1}, b_1 , ...
1
vote
2answers
49 views

Show that it doesn't exist any of natural number $ n = 4m + 3$ that $ n= x^2+y^2 $ for any natural x and y [duplicate]

Show that it doesn't exist any of natural number $ n = 4m + 3$ that $ n= x^2+y^2 $ for any natural x and y Show that every prime number in form $ p=4m+1 $ could be showed as $ p = x^2+y^2$ (x and y ...
9
votes
1answer
239 views

Why is $(\sqrt{2}+\sqrt{3})^{2008}$ so close to an integer?

Using 5000-digit precision in PARI/GP, I discovered that the fractional part of $(\sqrt{2}+\sqrt{3})^{2008}$ is extremely small, less than $10^{-999}$. Is there a simple explanation for this fact ? ...
2
votes
2answers
54 views

Transcendentals as the Roots of Infinite Polynomials

I have always been taught that the difference between an algebraic and a transcendental number is that the former is the root to a polynomial of ${\bf finite}$ degree with integer coefficients. I did ...
5
votes
1answer
89 views

Gentle introduction to algebraic number theory

For context, I am a undergraduate majoring in math. I've taken two semesters of algebra (though I am still a bit shaky on Galois theory). I just finshed a course in elementary number theory which used ...
1
vote
4answers
159 views

Solutions to the diophantine equation: $2a^2 + 2b^2- c^2- d^2 = 0$

As suggested on Mathoverflow (http://mathoverflow.net/questions/168536/solutions-to-the-diophantine-equation-2a2-2b2-c2-d2-0) I am transfering this question to math-stackexchange: I am looking for ...
2
votes
1answer
35 views

Ideals in a Quadratic Number Field

Show the ideal $I=\langle4,2+2\sqrt{-29}\rangle$ in $\mathbb{Z}[\sqrt{-29}]$ satisfies the equality $\langle8\rangle=I^{2}$ of ideals in $\mathbb{Z}[\sqrt{-29}]$. I tried to factorise $x^{2}+29$ over ...