The algebraic-number-theory tag has no wiki summary.

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### Linear independence of the square roots over Q

Does there exist a real number $a$ such that the numbers $\sqrt{n^2 + a^2}$ (for all natural $n$) are linearly independent over the field of rational numbers? It is evident that $a$ cannot be ...

**1**

vote

**1**answer

170 views

### Compatibility of two definitions of elliptic elements in GLn

For an element $g$ of a connected reductive group $G$ (over a local field),
$g$ is called $elliptic$ if it is semisimple and the maximal split subtorus of the center of the centralizer of $g$ is ...

**2**

votes

**1**answer

112 views

### Is any/every order on a number field forced by some finite extension?

Say a field extension $E/F$ forces the order on an ordered $F$ if every positive $x$ in $F$ is a sum of squares in $E$. A real closure of $F$ does this. And $\mathbb{Q}$ forces its own sole ...

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**6**answers

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### Text for Algebraic Number Theory

I have the privilege of teaching an algebraic number theory course next fall, a rare treat for an algebraic topologist, and have been pondering the choice of text. The students will know some ...

**4**

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**2**answers

217 views

### Even unimodular lattices with root system $32 A_1$

I'm studying Venkov's proof of the classification of even unimodular rank 24 lattices, and it prompted the following question.
For an even unimodular lattice $L$, let $R(L)= \{ x \in L : (x,x) =2\}$ ...

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**0**answers

310 views

### Does Hasse-Minkowski help to produce nontrivial rational solutions?

Consider a quadratic form over $\mathbb{Q}$, say, a diagonal one in three variables
$$
F(X, Y,Z) = a · X^2 + b · Y^2 − c · Z^2
$$
with positive integers $a,b,c$. Then $F(X,Y,Z)=0$ has a non-trivial ...

**2**

votes

**1**answer

245 views

### Cyclotomic integers with given modulus

The following problem was posted to the NMBRTHRY mailing list about a week ago, without eventually getting a satisfactory solution.
Suppose that $p=(n^2+1)/2$ is a prime, with $n\ge 5$ integer. Does ...

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votes

**2**answers

157 views

### Domains $D$ for which for any prime $P$, $D_P$ is a PID

Is there any name or alternative characterization for the class of integral domains $D$ such that for any prime ideal $P$, $D_P$ is a principal ideal domain?

**2**

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**1**answer

153 views

### Frohman & Fine's proof about Bianchi groups as HNN extensions (or anyone else's)

Specifically, I am looking for a proof that for squarefree $d\in\mathbb{N}\setminus\{ 1,3\}$, there exists some Kleinian $\Gamma$ and some $\alpha\in$Aut$(PSL(2,\mathbb{Z}))$, such that the Bianchi ...

**2**

votes

**1**answer

263 views

### Mordell-Weil and finiteness of rational points

Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order, that is, $\mathrm{End}_K(E)\cong \mathcal{O}_K$. Suppose the class number of $K$ is equal to $1$. Let ...

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vote

**1**answer

128 views

### Definition of level N congruence subgroup of an arithmetic group, useful for computations

My title requests something more general than I actually require right now, so I would settle for an answer to something more specific (details below) but I would like to understand the more general ...

**3**

votes

**1**answer

301 views

### Unramified extension and class field theory

I am not sure this question is proper for this site, but there is no other places that I can get an answer. So if anyone can give an answer for this, it would be very helpful to me.
Let $F$ be a ...

**0**

votes

**0**answers

224 views

### Understanding Umemura's Theorem for roots of algebraic equations

I am trying to understand Umemura's Theorem for expressing the roots of any algebraic equation by higher genus theta functions. The original paper can be found here: ...

**1**

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**0**answers

203 views

### Some identities with the Riemann-Hurwitz zeta function

The only definition that I have ever seen of this Riemann-Hurtwitz zeta-function is this,
For $0 < a \leq 1$ we have the identity
$$ \zeta(z, a) = \frac{2 \Gamma(1 - z)}{(2 \pi)^{1-z}} \left[\sin ...

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**2**answers

436 views

### Irreducible polynomials with a root modulo almost all primes

Let $f \in \mathbb{Z}[x]$ be a non-zero polynomial which is irreducible over $\mathbb{Q}$. Suppose that $f$ has a root in $\mathbb{F}_p$ for almost all primes $p$. Must $f$ be linear?
...

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**2**answers

506 views

### Hurwitz integers represented as sums of two squares of Hurwitz integers

I wonder if there exists a characterisation of Hurwitz integers which are represented as sums of two squares of Hurwitz integers, up to multiplication by a unit. And if so, could you please point to a ...

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**0**answers

317 views

### maximal abelian extension of quadratic extension of $\mathbb Q_p$

I read this article "Local class field theory via Lubin-Tate theory" http://arxiv.org/pdf/math/0606108v2.pdf. And I want to find the maximal abelian extensions for quadratic extensions of $\mathbb ...

**-4**

votes

**1**answer

147 views

### $p=4x^2+27y^2$,with $p$ a prime [closed]

p is a prime ,on what condition the Diophantine equation is solvable.what is it Linear expression ,for example ,$x^2+3y^2=p$, $p=3k+1$ ,$x^2+5y^2=p$ ,
$p=1,9\pmod{20}$.

**2**

votes

**1**answer

216 views

### The existence of elliptic curves with prescribed supersingular primes

For a given infinite set of primes, not too big, eg, satisfying Lang-Trotter conjecture, can we always find an E.C. with supersingular reduction (at least) at these primes? How about E.C. without CM?
...

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**2**answers

351 views

### The existence of infinitely many supersingular primes for every elliptic curve over Q

Elkies proved The existence of infinitely many supersingular primes for every elliptic curve over Q. I read his paper, but found the supersingular primes he constructed are all 3(mod 4) type. So, how ...

**3**

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**0**answers

152 views

### Is Eisenstein's map from binary cubic forms really injective when $\Delta > 0$?

The four binary cubic forms, homogeneous cubics of the form $C_j(x,y) = a x^3 + b x^2 y + c x y^2+d y^3 = (a,b,c,d)$,
$C_1 = (1,3,-15,-23)$, $C_2 = (1,3,-51,-203)$, $C_3 = (1,3,-33,-113)$, $C_4 = ...

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votes

**1**answer

690 views

### Growth of the “denominator” of powers of an algebraic number

Let $z \in \overline{\bf Q}$ be an algebraic number. Define the "denominator" of z to be the least natural number $n$ such that $nz$ is an algebraic integer.
By a rather ad hoc argument (playing ...

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**0**answers

82 views

### Constructing a polyhedron of maximal possible volume from given bounds on areas of its faces

Consider $n$ variables $a_1,...,a_n$ ranging over $\mathbb{R}^+$. Suppose we are given $n$ pairs of positive rational numbers $(p_1,q_1),...,(p_n,q_n)$ where each pair imposes bounds on the ...

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votes

**1**answer

452 views

### About principal ideal theorem in number fields

I usually consider a cyclic extension $K$ of degree an odd prime $p$ over the rational field $\mathbf{Q}$.
In this case, there is a well-known result that "every ambiguous class in the class group ...

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votes

**2**answers

670 views

### Cohen-Lenstra Heuristics reference

I am looking for good references (preferably, books) on Cohen-Lenstra Heuristics (on Real Quadratic fields) which explain in detail the reasons behind its fundamental assumption (higher the ...

**5**

votes

**1**answer

373 views

### Minimal polynomial of sums of roots of unity with constant term $\pm1$

Let prime $p$ and given $\zeta_p = e^{2\pi i/p}$. It is well-known that the minimal polynomial of $x = \zeta_p + \zeta_p^{p-1}$ has a constant term either $\pm 1$ and, for certain $p$, the sum of ...

**1**

vote

**1**answer

148 views

### Algebraic numbers abhorrent to cyclotomic fields

Consider an algebraic number $\alpha$, which can be taken to be an
integer. With $\deg\alpha$ a prime number, one can easily arrange that
to be such that all powers $\alpha^n$ to be of the same ...

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**1**answer

336 views

### When can number rings be spanned (as $\mathbb{Z}$-modules) by units?

Let $\mathcal{O}$ be the ring of integers in an algebraic number field. Define $R \subset \mathcal{O}$ to be the set of all $\mathbb{Z}$-linear combinations of units. Since the product of two units ...

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**1**answer

193 views

### A characterization of the module function on a locally compact division ring

The same question was asked in Math StackExchange about 3 months ago.
Since nobody has answered to it, I would like to post it here.
References:
Weil's Basic Number Theory(denoted by BNT).
...

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**1**answer

473 views

### The Dissertation of F. J. van der Linden

Does anyone have access to the 1984 dissertation of Franciscus Jozef van der Linden under Hendrik Lenstra? It is called Euclidean Rings with two infinite primes. The theory is that this has the ...

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216 views

### Euclidean real quadratic fields

It is known that, under GRH, a real quadratic field is Euclidean iff it is a UFD. So, assuming the conjecture of Gauss and GRH, we expect that there are infinitely many Euclidean real quadratic ...

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**1**answer

264 views

### Number Field Sieve for factorization with non-monic non-linear polynomial. Can't understand calculating ideal valuations

There is a paper "Factoring integers with the number field sieve" (download it here, for example).
I can't understand how they reason the correctness of computing ideal valuations in the case of ...

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**1**answer

509 views

### primes represented by an indefinite binary quadratic form

Suppose I have a form $$ f(x,y) = a x^2 + b x y + c y^2, $$ with $a,b,c$ integers, $\gcd(a,b,c)=1$ and $\Delta = b^2 - 4 a c > 0,$ but $\Delta \neq n^2$ for any integer $n.$
Do there exist ...

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**0**answers

204 views

### Generating congruence subgroups of SL_n over totally imaginary number rings

Fix some $n \geq 3$. Let $k$ be an algebraic number field with ring of integers $\mathcal{O}$ and let $\alpha$ be an ideal of $\mathcal{O}$. Define $\text{SL}_n(\mathcal{O},\alpha)$ to be the ...

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**1**answer

298 views

### Some puzzles about the three conditions in a paper of D.Berend

Recently, I am reading a paper titled "multi-invariant sets on tori" by D.Berend.
I am puzzled by the three necessary and sufficient conditions given there.
Could you provide me with some concrete ...

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**0**answers

175 views

### When is a Number Ring generated by its Norm-1 elements?

In exercise 1.2.11. of Bump's Automorphic Forms and Representations book, he deals with real quadratic fields $K$ in which $\mathcal{O}_K$ is generated (as a ring) by its norm-1 units. In this case ...

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votes

**1**answer

61 views

### Action of GL(2,O_k) on 1d subspaces of (O_k)^2

Let $\mathcal{O}_k$ be the ring of integers in an algebraic number field $k$. Let $M$ be a rank $1$ projective module over $\mathcal{O}_k$ (in other words, $M$ is a projective module such that $k ...

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votes

**1**answer

570 views

### Prime splitting in cubic field, congruence [closed]

Let $K$ be a cubic Galois extension of $\mathbb{Q}$.
I wonder if we can find a congruence for prime $p$ such that $p$ does not split completely in $K$. I know that we can do this for quadratic ...

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votes

**1**answer

249 views

### Reduction mod $p$ of units in a ring of integers

Let $\mathcal{O}_k$ be the ring of integers in an algebraic number field $k$ and let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_k$. I'm looking for conditions on $k$ and $\mathfrak{p}$ which ...

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**0**answers

162 views

### Lang's preprint “Cyclotomic points, very anti-canonical varieties, and quasi-algebraic closure”

I am trying to find the following preprint of Serge Lang, which supposedly discusses his C1 conjecture:
"Cyclotomic points, very anti-canonical varieties, and quasi-algebraic closure".
I have not ...

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**1**answer

249 views

### Rational points of non-rational curves

An algebraic curve (in this question) is the zero set $C = f^{-1}(X\ Y)$ of any polynomial $f\in\mathbb R[X\ Y]$; we say then that $f$ represents $C$. ...

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**2**answers

368 views

### property of trace modulo $n$

I recently noticed an interesting (at least to me) property of the trace but have been unable to prove it.
Let ${\mathbb K}$ is an algebraic number field with ${\mathcal O}$ as its ring of integers, ...

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**2**answers

276 views

### Question about ring of integers of cyclotomic field [closed]

Let $\zeta=e^{\frac{2\pi i}{p^n}}$, $p$ is an odd prime. Is $\mathbb{Z}[\zeta]$ a UFD?
Thank you for watching.

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**1**answer

194 views

### RefReq: Algorithms for standard operations in Algebraic Number theory

Given an algebraic number field $F$ (I actually don't have an idea how to implement this data already, except for splitting fields of polynomials, but there is something in SAGE) is there free code ...

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204 views

### Salem and Perron polynomials

If $P(t)\in \mathbb{Z}[t]$ is a polynomial, let $d$ be its degree and let $P_{*}(t)$ denote its reciprocal polynomial, i.e. $P_{*}(t) := t^d\, P(1/t)$.
Let $Q_n(t) \in \mathbb{Z}[t]$ be a polynomial ...

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**1**answer

604 views

### An integral representation of the Riemann zeta function

I am referring to the equality in equation $3.29$ (page 12) and $4.20$ (page 17) in this paper.
I am unable to recognize where this comes from or what is the general expression for values other than ...

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97 views

### Derivative of a function related to Dedekind zeta function

Lef $K$ be an algebraic number field of degree $[K:\mathbb{Q}]=n$. For simplicity suppose $K$ is totally real. Define $f(s) = \zeta_K(s) \zeta(1-s)^{n-1}$ where $\zeta = \zeta_{\mathbb{Q}}$.
From the ...

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260 views

### Inequalities in paper by Jean Bourgain

The question refers to the following paper by Jean Bourgain: http://arxiv.org/abs/math-ph/0011053
Specifically, I can't derive the following inequality in (1.20):
\begin{equation}
...

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127 views

### Diophantine approximations by norms of quadratic irrrationalities

The following problem came up on a mailing list that I subscribe to:
If $\alpha$ is irrational we can find (using continued fractions) infinitely many rational fractions $p/q$ such that $|q \alpha - ...

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**0**answers

110 views

### Isogenies in multidimensional formal groups

Let $K/\mathbb{Q}_p$ be a local field, $A$ the ring of integers of K, $\pi$ a uniformizer element for $A$, $F$ an n-dimensional formal group with coefficients in $A$ and $f$ an endomorphism of $F$. ...