**1**

vote

**0**answers

122 views

### What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$? [on hold]

Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$.
If $L$ is a number field ...

**0**

votes

**0**answers

102 views

### More generalized RSA construction

Is there a way to construct RSA type cryptosystem over general number rings?
Can Number Field Sieve technique be applied here?

**3**

votes

**0**answers

57 views

### An order in $\mathbb Q[G]$ which is a maximal $\mathbb Z_p$-order in $\mathbb Q_p[G]$ for finitely many primes $p$

Let $G$ be a finite group and $S$ a finite set of prime numbers. I know that every separable $\mathbb Q$-algebra $A$ contains a maximal $\mathbb Z$-order but I wonder if the following is true.
Is ...

**5**

votes

**2**answers

158 views

### Mahler measure of a totally positive, expanding algebraic integer

Consider a degree-$d$ algebraic integer $\alpha$ all of whose conjugates (including itself) are real numbers greater than 1. Its Mahler measure $M(\alpha)$ is simply equal to the norm $N(\alpha)$. ...

**4**

votes

**2**answers

261 views

### The best possible density in Hilbert's Irreducibility Theorem

Let $f(X,t_1,\dots,t_s)$ be an irreducible polynomial with coefficients in $\mathcal{O}_K$, the ring of integers of a number field $K$. By work of S. D. Cohen ...

**7**

votes

**1**answer

193 views

### If two Hecke characters cut out the same field, are they Galois conjugates?

First question on MathOverflow, I hope it is appropriate for this site. There are two related questions.
Let $K$ be a number field, $G_K = Gal(\overline{K}/K)$, $p$ a prime, and
...

**5**

votes

**1**answer

305 views

### Unique quadratic subextension of a ray class field

Let $K_q$ denote the unique quadratic subextension of the ray class field over $\mathbb{Q}$ of conductor $q\times\infty$. Then $K_q$ should be $\mathbb{Q}(\sqrt{q})$ if $q$ if 1 mod 4 and ...

**15**

votes

**4**answers

2k views

### Is there an elementary way to find the integer solutions to $x^2-y^3=1$?

I gave this problem to my undergraduate assistant, as I saw that Euler had originally solved it (although I am having trouble finding his proof). After working on it for two weeks, we boiled the hard ...

**11**

votes

**0**answers

478 views

### What is known about the reverse mathematics of algebraic number fields?

I know work on the reverse mathematics of countable algebraic field extensions including Galois theory, notably including Dorais, Hirst, and Shafer http://arxiv.org/pdf/1209.4944v2.pdf. But algebraic ...

**9**

votes

**0**answers

445 views

### Algebraic proofs of algebraic theorems about algebraically closed fields

It is well-known that the first order theory of algebraically closed fields admits quantifier elimination, whence the theory $ACF_p$ of algebraically closed fields of given characteristic $p$ is ...

**3**

votes

**0**answers

107 views

### How to Taylor series expand at the prime at infinity

Given a rational number, one can find a Taylor series expansion with respect to any $p$-adic valuation, as covered in Gouvea's introductory text on $p$-adic numbers. My question is how does one do ...

**12**

votes

**1**answer

204 views

### Are quadratic units cyclotomic norms?

Consider the fundamental unit $\varepsilon$ of a real quadratic number field $k = {\mathbb Q}(\sqrt{p})$ for primes $p \equiv 1 \bmod 4$, and let $h$ denote its class number. By Dirichlet's work on ...

**12**

votes

**5**answers

1k views

### How much do I need to learn algebraic geometry to understand arithmetics over number fields

I am at the stage of learning. Mostly, I am attracted by algebraic number theory. Roughly speaking, I am interested in the rational points of algebraic varieties. I am little bit afraid to start to ...

**3**

votes

**0**answers

112 views

### How to show that $h(-D)\geq \displaystyle\sum_{a\in A_1\\, 1\leq a\leq{\frac{\sqrt D}{2}}} 1$?

Here $A_1=\{u;p|u\Longrightarrow \chi(p)=1\}$ with $\chi$ a real quadratic character and $h(-D)$ the class number of the imaginary quadratic field of the fundamental discriminant. This problem occurs ...

**11**

votes

**3**answers

542 views

### Philosophy behind cohomological representations

For a given real reductive Lie group $G$, we have the notion of a representation being cohomological using the Lie algebra cohomology. In particular we know that the discrete series representations of ...

**3**

votes

**1**answer

361 views

### On the Diophantine equation $x^2 = y^p + 2^{r}z^p$ where $p\geq 7$ is an odd prime and $r \geq 2$

It is known that the only nonzero pairwise coprime integer solutions to the above Diophantine equation are for $r=3$, for which $(x, y, z) = (3,1,1)$ and $(-3,1, 1)$. (Cohen, Number Theory Volume 2: ...

**12**

votes

**3**answers

645 views

### Profinite groups as absolute Galois groups

It is a well-known result that all profinite groups arise as the Galois group of some field extension.
What profinite groups are the absolute Galois group
$\mathrm{Gal}(\overline{K}|K)$ of some ...

**14**

votes

**1**answer

659 views

### Even Galois representations “mod p”

Consider an irreducible $\mathrm{mod}$ $p$ representation:
$$\rho: \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2(\bar{\mathbb{F}}_p)$$
If $\rho$ is odd, it was conjectured by Serre in ...

**2**

votes

**1**answer

267 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 ...

**24**

votes

**10**answers

3k views

### What are some interesting problems in the intersection of Algebraic Number Theory and Algebraic Topology?

I'm a beginning graduate student and while my background is primarily in algebraic number theory, I've found myself a bit smitten with the subject of algebraic topology recently after only having read ...

**16**

votes

**2**answers

719 views

### Special topics to include in course in algebraic number theory

I'll be teaching an introductory course in algebraic number theory this fall (stopping before class field theory). I'm looking for a good list of "special topics" I can include to illustrate the ...

**23**

votes

**1**answer

470 views

### Artin reciprocity $\implies $ Cubic reciprocity

I asked this on math.SE a few days ago with no reply, so I'm reposting it here. Hope this is not considered too elementary for MO (feel free to close if so).
I'm trying to understand the proof of ...

**-1**

votes

**1**answer

61 views

### Equation with norms of cyclic extensions of coprime degrees

Let $\mathbb{K}$ be a quadratic extension of $\mathbb{Q}$ and $\mathbb{L}$ be a cyclic extension of $\mathbb{Q}$ of odd degree. Given a rational $r\neq 0$, does there always exist $k\in \mathbb{K}^*$ ...

**3**

votes

**0**answers

98 views

### Siegel's article “The volume of the fundamental domain for some infinite groups”: trouble with understanding computations

This is the article I mentioned. While the idea of what Siegel is doing in order to compute the volume of the fundamental domain described in the article (the very first one, for there are discussed ...

**3**

votes

**0**answers

144 views

### Computing algebraic properties of trace fields, as given by SnapPy

SnapPy can tell you the trace field of a hyperbolic $3$-manifold (which is awesome), but it specifies the field by outputting:
the minimal polynomial of the field over $\mathbb{Q}$, and
a decimal ...

**12**

votes

**1**answer

207 views

### A prime ideal $\mathfrak{p}$ decomposes in $\mathbb{Q}(\zeta_{24})/\mathbb{Q}(\sqrt{-6})$ iff it is generated by $\alpha\in1+2\Bbb{Z}[\sqrt{-6}]$

For a nonzero prime ideal $\mathfrak{p}$ of $\mathbb{Z}[\sqrt{-6}]$ which does not divide $2$, does $\mathfrak{p}$ decompose completely in the extension $\mathbb{Q}(\zeta_{24})/\mathbb{Q}(\sqrt{-6})$ ...

**3**

votes

**0**answers

72 views

### Sign of bivariate polynomial evaluated over two algebraic numbers

I would like to compute the sign of a bivariate polynomial $f$ evaluated over two algebraic numbers $a$, $b$. The numbers are in "isolating interval representation" meaning that each one is defined by ...

**4**

votes

**0**answers

119 views

### extending $p$-adic character of the local intertia to the absolute Galois group

Suppose I have a number field $F$, and a finite place $v$ of $F$. Let $E$ be finite extension of $F_v$. I start with a continuous morphism
$$
\chi \colon O_{F_v}^\times \to E^\times.
$$
where ...

**35**

votes

**13**answers

3k views

### Applications of the Cayley-Hamilton theorem

The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own ...

**6**

votes

**3**answers

254 views

### For an arithmetic hyperbolic 3-manifold group, when is its trace field not its invariant trace field?

Edit: In my original post I failed to require the group to be a manifold group. The answer below from @BenLinowitz works in that case. I am really interested though in when the group is torsion-free, ...

**6**

votes

**2**answers

325 views

### Are the abelian absolute Galois groups of these local fields isomorphic?

For a field $F$ we denote by $F^{\mathrm{ab}}$ the compositum of all finite Galois abelian extensions of $F$.
Is $\mathrm{Gal}(\mathbb{Q}_2(\sqrt[8]{3})^{\mathrm{ab}}/\mathbb{Q}_2(\sqrt[8]{3})) ...

**18**

votes

**1**answer

691 views

### Concrete Applications of knowing $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$

I have very little experience with Galois representations, mostly as they relate to class field theory, elliptic curves, and modular forms, but they seem to have quite a reputation in number theory as ...

**10**

votes

**1**answer

375 views

### What was a cusp to Hurwitz in 1892?

Let $d\in\mathbb{N}$
be squarefree.
Let $\mathcal{O}_d$
be the ring of integers of $\mathbb{Q}(\sqrt{-d})$.
Let $\Gamma_d=\mathrm{PSL}_2(\mathcal{O}_d)$.
Let $\mathcal{H}^3$
be the upper half-space ...

**5**

votes

**0**answers

107 views

### Factorization problem in Cyclic cubic field

Let K/$\mathbb{Q}$ be a cubic number field. Assume that K/Q be Galois with class number 1.
Therefore Gal(K/Q) is cyclic cubic group and $\mathcal{O}_K$ is a PID.
Let p be a rational prime, p ...

**10**

votes

**2**answers

2k views

### Why is $\frac{\sqrt{6}}{32}(29 + \sqrt{145}) \approx \pi$ ?

Apologies in advance if this is a stupid question; also, disclaimer: this is purely for fun; but:
Why is $\frac{\sqrt{6}}{32}(29 + \sqrt{145})$ such a good approximation to $\pi$?
(Correct to 8 ...

**0**

votes

**1**answer

52 views

### Norm Residue Symbol refinement?

From Wikipedia: given $a\in K^\times$,
(a,b)=1 for all b [in K*] if and only if a is in K*ⁿ
So suppose that $(\frac{a\ ,\ K^\times\!}{p})\neq 1$ [assume $n$ above ...

**46**

votes

**13**answers

5k views

### Erratum for Cassels-Froehlich

Edit 25 April 2010: I have a physical copy of the new printing of the book. I can only assume the LMS is now selling it (but have no details).
IMPORTANT EDIT: THE RESULTS ARE IN! Ok, the deadline has ...

**15**

votes

**1**answer

359 views

### Is the number of representations as the sum of two elements of a polynomial sequence always small?

Let $f(x) \in \mathbb{Z}[x]$ be a degree $d>1$ polynomial with integer coefficients. Define
$$r(n) := | \{x,y \in \mathbb{Z} : f(x)+f(y) = n \}|. $$
My question is:
Is it true that ...

**9**

votes

**1**answer

294 views

### Parametrizing all cyclic extensions of the rational numbers of degree 5

Is there a polynomial $f(T,X) \in \mathbb{Q}(T)[X]$ in the indeterminate $X$ over the field $\mathbb{Q}(T)$ with $\mathrm{Gal}(f/\mathbb{Q}(T)) \cong \mathbb{Z}/5\mathbb{Z}$ such that for every Galois ...

**14**

votes

**1**answer

468 views

### Growth of $\zeta_{\mathbf Q[\cos(\frac{\pi}{2^{n+1}})]}(2)$

Let $K_n$ be the field $\mathbf Q[\cos(\frac{\pi}{2^{n+1}})]$ (the real subfield of the cyclotomic field $\mathbf Q[e^{\frac{i\pi}{2^{n+1}}}]$).
Is there anything known about the growth of the ...

**1**

vote

**1**answer

112 views

### Existence of class modules for finite groups

I asked the following question on Stackexchange and got no reply so I am reposting it here. Let $G$ be a finite group. A $G$-module C is a class module if, for all subgroups $H \subset G$:
1) ...

**16**

votes

**2**answers

832 views

### Can the Dedekind zeta function distinguish between real and imaginary quadratic number fields?

Suppose I am given a machine that gives me the coefficients $a_1$, $a_2$, $a_3$, ... of a Dirichlet series
$$\sum_1^{\infty} \frac{a_n}{n^s} $$
and assume that I know that this Dirichlet series is the ...

**4**

votes

**1**answer

157 views

### What is the explicit eigenvalues of Hilbert modular forms?

Let $F$ be a totally real number field and let $I$ denote the set of embeddings
$\tau:F\to \mathbb{R}.$ Let $k=(k_\tau)\in\mathbb{Z}^I_{>0}$ and suppose all the $k_\tau$'s have the same parity. Let ...

**8**

votes

**4**answers

662 views

### j-invariant fixed point?

If we view the j-invariant of a lattice as a map from the upper-half plane to the complexes by $\tau\mapsto j([1,\tau])$, then it is surjective, holomorphic, and has quite a number of other wonderful ...

**4**

votes

**1**answer

177 views

### What can we say about the differences between roots of a polynomial with large Galois group?

Suppose that $K$ is a number field and $L$ is the splitting field of a monic polynomial in $\mathcal{O}_{K}[x]$ of degree $d \geq 5$ with roots $\alpha_{1}, ... , \alpha_{d}$. Assume that the ...

**18**

votes

**2**answers

654 views

### References for $K_{4k}(\mathbb{Z})$

Weibel's "Algebraic K-theory of rings of integers in local and global fields" says $K_{4k}(\mathbb{Z})$ are known to have odd order, with no prime factors less than $10^7$, but are conjectured to be ...

**0**

votes

**0**answers

48 views

### Kernel of the Artin map when dealing with S-ideles and S-divisors for function fields

Having understood that there is a strong correspondence between number fields and function fields, I am trying to work out some function field equivalents of class field theoretic invariants from Bost ...

**1**

vote

**0**answers

111 views

### Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers upto $\frac{1}{2}p^2_x -2$ exvert all non-trivial $\Xi_x$-classes? [closed]

This question follows from the information provided below.
Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers up to $\frac{1}{2}p^2_x -2$ exvert all non-trivial ...

**22**

votes

**2**answers

778 views

### Elementary congruences and L-functions

In a recent article, Emmanuel Lecouturier proves a generalization of the following surprising result: for a Mersenne prime $N = 2^p - 1 \ge 31$, the element
$$ S = \prod_{k=1}^{\frac{N-1}2} k^k $$
...

**6**

votes

**1**answer

177 views

### Property of Dirichlet character

Let $\chi_0$ be the unique Dirichlet character $\text{mod }1$ (i.e. $\chi_0(n) = 1$ for all $n$), $\zeta_p$ be a primitive $p$th root of $1$, and for any $a \in \mathbb{F}_p^\times$ and any Dirichlet ...