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183 views

### Easiest way to see that $\zeta_{\mathbb{Z}[i]}(s) = \zeta(s) L(s, \chi)$?

As the question suggests, what is the easiest way to see that$$\zeta_{\mathbb{Z}[i]}(s) = \zeta(s)L(s, \chi)?$$Here, $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to \mathbb{C}^\times$ ...

**4**

votes

**1**answer

71 views

### Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$, does $L_c(s, \chi)$ necessarily equal $1$?

Consider an analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$. Let $g \in \mathbb{F}_q[T]$, $g \neq 0$, let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a homomorphism, let $c ...

**6**

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87 views

### $F[[T]] \times F[[1/T]]$ fundamental domain, show compactness

Let $p$ be a prime number. What is the easiest way to see that $(\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T)))/\mathbb{F}_p[T, 1/T]$ is compact? Here $\mathbb{F}_p[T, 1/T]$ is embedded in ...

**1**

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50 views

### Can we prove the uniqueness of the local Artin map by using mostly global class field theory?

Let $l/k$ be a finite abelian extension of $p$-adic fields. There is a well defined local Artin map $k^{\ast} \rightarrow Gal(l/k)$ with kernel $N_{l/k}(l^{\ast})$. Let's suppose that we have only ...

**-1**

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**0**answers

49 views

### square classes of quadratic extensions of 2-adic fialds [migrated]

I have a question about square classes of quadratic extensions of 2-adic fields. I appreciate anybody help me to understand.
Why all elements of $1+\mathfrak{p}^5$ are square in ...

**3**

votes

**1**answer

156 views

### A question about “small” sets of algebraic numbers

For the sake of brevity let me use the following terms. A subset $X$ of $\bar{\mathbb{Q}}$ will be called "small" if for any number field $K$, the intersection
$X\cap K$ is finite. Similarly, a set ...

**3**

votes

**1**answer

432 views

### Is this polynomial irreducible over the rationals?

Prove (or disprove): Define $T_n(x)$ as the Chebyshev polynomial of the first kind with degree $n$ . If $p$ is an odd prime, then $\sqrt{\frac{T_p(x)-1}{x-1}}$ is an irreducible polynomial over the ...

**0**

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**0**answers

63 views

### root of unity in a local field [migrated]

Let $K$ be a local field with residue field of cardinality $q$ that contains the $n$-th roots of unity (for some $n\in\mathbb N^*$). Does $n$ divide $q-1$?
If not, is there a relation between $n$ and ...

**0**

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**0**answers

39 views

### counting fundamental units of real quadratic fields

For a given real quadratic field $ K$, the units of its ring of integers is $\mathcal{O}_K\cong(\pm1)\times \mathbb{Z}$ by Dirichlet unit theorem. For each $\mathcal{O}_K$, pick the fundamental unit ...

**5**

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102 views

### Furtwangler's Principal ideal theorem in number fields

Does anyone know a simple proof, using cohomological method of the fact that the verlagerung from a finite group G. to its commutator subgroup G', i.e. $$G/G'->(G')^{ab}$$ vanishes?
The simplest ...

**2**

votes

**0**answers

70 views

### $K^{ur}K^{\pi} = L$

Let $K$ be a $p$-adic field, and $L$ an infinite abelian extension of $K$ containing $K^{ur}$. Let $\Phi: K^{\ast} \rightarrow Gal(L/K)$ be the local Artin map. Let $\pi$ be a uniformizer for $K$, ...

**4**

votes

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120 views

### Is $K^{ur} K^{\pi} = L$?

Let $L/K$ be a finite extension of $p$-adic fields, $\pi$ a uniformizer of $K$, $\theta = (-, L/K)$ the local Artin map $K^{\ast} \rightarrow Gal(L/K)$, $E$ be maximal unramified extension of $K$ ...

**6**

votes

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68 views

### Minimal Discriminants

Let $D_n$ be the minimal absolute value of the discriminants of
number fields with degree $n$. Arnold Scholz conjectured in 1936 that
$D_{397} > D_{400}$, which is, of course, still open (Scholz ...

**14**

votes

**1**answer

244 views

### Geometric Mean of $L(1,\chi)$ for quadratic Dirichlet characters

Let $S = \{D_1, D_2, D_3, \ldots \}$ be the set of all prime discriminants
(or positive prime discriminants) of quadratic number fields. For such a
discriminant let $\chi_j(n) = (\frac{D_j}n)$ be ...

**3**

votes

**2**answers

238 views

### N-th root of unity in N-th division field of abelian variety?

Let K be a number field and A/K an abelian variety over it. Can it be that K(A[n]) does not contain a primitive n-th rooth of unity? If the answer is yes is it always possible to bound the bad n ...

**0**

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**0**answers

36 views

### Ambiguous classes not strongly ambiguous

I am looking for a quadratic field $K$ and a class $\mathcal C$ of the classes group of $K$ such that $\mathcal C$ is ambiguous but not strongly ambiguous that is:
$\sigma(\mathcal C)=\mathcal C$ for ...

**4**

votes

**0**answers

63 views

### Explicit extensions for Heisenberg groups

Let $G$ be the $p$-adic Heisenberg group $\begin{pmatrix} 1&\mathbb Z_p&\mathbb Z_p\\&1&\mathbb Z_p\\&&1\end{pmatrix}$. Is it possible to write an explicit extension $K/k$, ...

**3**

votes

**2**answers

224 views

### A galois group is topologically finitely generated or finitely generated as a $\mathbb{Z}_p$-module

The basic set up is the following:
Let $K$ be a number field. let $p$ be an odd prime. Let $\Sigma_p$ be the set of primes of $K$ lying above $p$. Let $M$ be the composite of all finite $p$-extensions ...

**16**

votes

**1**answer

413 views

### Quickest and/or most elementary proof of “principal iff splits completely”?

Let $L$ be the Hilbert class field of a number field $K$, and let $\mathfrak{p}$ be a prime ideal of $K$. Then $\mathfrak{p}$ splits completely in $L$ if and only if $\mathfrak{p}$ is a principal ...

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33 views

### Which properties determine the uniqueness of the local Artin map?

Any abelian extension of local fields can be realized as the completion of a global abelian extension. So let $L/K$ be abelian, $w/v$ an extension of places. From the global Artin map on ideles we ...

**2**

votes

**1**answer

111 views

### References about identities of Gauss sum

I am reading the paper. In the end of page 10, there are the following identities of Gauss sum.
\begin{align}
& h(b) h(a+b) = q^b h(b) h(a), \\
& h(b) g(a+b) = q^b h(b) g(a), \\
& g(a+b) ...

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votes

**2**answers

123 views

### Is any F-stable maximal torus contained in some F-stable Borel subgroup? [closed]

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...

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83 views

### Surjectivity of ring homomophism induced by Frobenius endomorphism

Denote by $F_q$ the finite field with $q$ elements, and denote by $\bar{F_q}$ its algebraic closure. Let $V$ be an affine $\bar{F_q}$-variety and $F$ be the Frobenius endomorphism corresponding to an ...

**3**

votes

**0**answers

150 views

### Looking for a copy of Algebraic Number Theory in honor of Iwasawa

I am looking for an electronic copy of this volume:
Advanced studies in Pure Mathematics, Volume 17
Algebraic Number Theory - in honor of K. Iwasawa
Edited by J. Coates, R. Greenberg, B. Mazur and I. ...

**1**

vote

**1**answer

238 views

### When can we write fundamental units explicitly

Given a number field $K$, the Dirichlet Unit Theorem tells us about the structure of the unit group $O_K^\times$. However, the proofs do not seems to give any way to explicitly write out a set of ...

**8**

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**1**answer

317 views

### Orthonormal bases of R^3 with components lying in the golden field

Greg Egan proved an interesting theorem about unit vectors in $\mathbb{R}^3$ whose components actually lie in the 'golden field' $\mathbb{Q}[\sqrt{5}]$. He found it in our studies of twin ...

**9**

votes

**2**answers

597 views

### What are the current trends in class field theory?

Being far from an expert in the subject I was wondering if people can hint towards a modern exposition of the developments in the last 10 years ? Or if not then suggest some sub-subjects in CFT that ...

**7**

votes

**1**answer

189 views

### Lower bound on class number of binary quadratic forms of discriminant of the form $n^2+4$

While searching for a use for the "sum invariant" of indefinite binary quadratic forms of discriminant $D = n^2 + 4$ (see https://cs.uwaterloo.ca/journals/JIS/VOL17/Smith/smith5.html), I believe I ...

**10**

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**1**answer

441 views

### Class field towers

It is known (Golod and Shafarevich) that the class field tower of a finite extension $K$ of $\mathbb{Q}$ may be infinite. But is it always finite for $K=\mathbb{Q}[\zeta]$ where $\zeta$ is a root of ...

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**2**answers

215 views

### Surjectivity of trace map

Let $R$ be a closed integral domain with its fraction field $F$. Let $K$ be a finite separable extension field of $F$, and let $A$ be the integral closure of $R$ in $K$.
It is well known that the ...

**13**

votes

**1**answer

470 views

### When complex conjugation lies in the center of a Galois group

Let $K \subseteq \mathbb{C}$ be a number field (I'm fixing an embedding), and assume $K/\mathbb{Q}$ is Galois with Galois group $G$. Let $\tau \in G$ denote complex conjugation. This question ...

**0**

votes

**1**answer

120 views

### Algebraic Hecke characters with a given infinite part

I'm needing to find out if there exists an algebraic Hecke character for a number field F, $\phi: \mathbb{A}_F \rightarrow \mathbb{C}$, for a fixed infinite part $\phi_\infty$ and a fixed component ...

**2**

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**0**answers

194 views

### Algebraic integer with conjugates on the unit circle

Let $\alpha$ be an algebraic integer on the unit circle in $\mathbb{C}$ such that all the conjugates of $\alpha$ lie on the unit circle. Does it follow that $\alpha$ is a root of unity?

**2**

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141 views

### Normal basis in cyclotomic number fields

Let $p$ be an odd prime integer and let $\zeta$ be a primitive $2p$-th root of unity. Does $\alpha=1+\zeta+\zeta^{-1}+\dots+\zeta^{\frac{p-1}{2}}+\zeta^{-\frac{p-1}{2}}$ generates a normal basis of ...

**6**

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**4**answers

516 views

### Some Non-Trivial Algebraic(Rational) Number

Every problem about algebraic-ness (rational-ness) of numbers that I have seen is in one of the below types:
The number is algebraic(rational) and proving that it is algebraic(rational) is trivial, ...

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**0**answers

105 views

### Extension of a complete discrete valuation ring

My question came when I was reading the famous Tate's paper on $p$-divisible groups. At the beginning of chapter $(2.4)$ he cites this fact as obvious. If you take a complete discrete valuation ring ...

**8**

votes

**2**answers

871 views

### divisible by all standard prime numbers

This question is about prime numbers in nonstandard models of Peano Arithmetic. Every such model looks like N+AxZ, where A is a dense linear order without end points.
There are many nonstandard ...

**0**

votes

**1**answer

162 views

### Normal basis with cyclotomic units

Let p be an odd prime integer and let $\zeta$ be a primitive p-th root of unity.
Let $\alpha$ be a non-trivial cyclotomic unit of $\mathbb Q(\zeta)$, i.e. an element of the form ...

**5**

votes

**1**answer

351 views

### A strange condition on containment of special complex numbers in cyclotomic fields

In a recent theorem we have naturally come across this condition, that seems to be important, but rarely satisfied:
$\sqrt{\frac 1 4 + a^m} \in \mathbb Q(\zeta_m, a)$
where $a\in\mathbb C^*$ and ...

**6**

votes

**1**answer

330 views

### UFD and fundamental group

Let $C$ be the curve $x^2+y^2-1$, defined over $\mathbb R$. It is easy to see that $\mathbb R[C]$ is not a UFD, as witnessed by the identity $(1-x)(1+x)=y^2$. On the other hand, the real locus ...

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106 views

### Solving the transcendental equation $Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3$

I need to solve the following equation:
$Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3$
for $x\in\mathbb{R}^{\ast}$ and where $k\in\mathbb{R}^{+}$. Here $Li_{3}$ and $Li_{2}$ are the third and ...

**3**

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187 views

### Density of primes of degree one in Bauer's Theorem (Application of Chebotarev Density)

Let $L$ be a Galois extension of $\mathbb{Q}$ and $M$ a finite extension of $\mathbb{Q}$, both of degrees $> 1$. A Theorem of Bauer tells that $Spl_1(M)\subset Spl(L)$ up to a finite number of ...

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416 views

### How to compute class number of a torus

Let $T$ be an algebraic torus over a number field $K$.
Following notations in Ono's The Arithmetic of Tori,
...

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108 views

### algorithm to find a new point of small height in a number field extension

By the height of an algebraic number $\alpha$, I mean the absolute, logarithmic (additive) Weil height $h(\alpha)$; e.g. $h(2^{1/n}) = (\log 2)/n.$
If $K$ is a number field, let $\delta(K)$ denote ...

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151 views

### Unramified extensions of a given degree

Let $K \neq \mathbb{Q} $ be a finite extension of $\mathbb{Q}$. For a given integer $n$, how to construct an unramified extension of $K$ of degree $n$ ?
EDIT: If not then under what conditions on ...

**0**

votes

**1**answer

96 views

### A particular argument in the review on expanders by Hoory-Linial-Wigderson

I am thinking about the third bullet point on page 455 here, http://www.ams.org/journals/bull/2006-43-04/S0273-0979-06-01126-8/
Can someone explain what is the argument there which seems to conclude ...

**7**

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**2**answers

278 views

### Constructing Ramanujan graphs from elliptic curves

Is there an exposition which explains how to do this step-by-step? (I see stray references and allusions to such a thing being possible but can't locate anything concretely)
Something to do with ...

**3**

votes

**1**answer

220 views

### Dihedral extension of 2-adic number field

Sorry if the question is too long and maybe elementary.
I am reading a paper by Hirotada Naito on "Dihedral extensions of degree 8 over the rational p-adic fields". To generate dihedral extension ...

**6**

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**0**answers

131 views

### Degenerate linear recurrence sequences

Let $(u_n)_{n \geq 0}$ be a linear recurrence given by
$$u_n = a_1 u_{n-1} + \cdots + a_k u_{n-k} \quad \forall n \geq k ,$$
where $u_0, \ldots, u_{k-1}, a_1, \ldots, a_k \in \mathbb{Z}$. We recall ...

**5**

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**1**answer

457 views

### Group laws in class field theory

In the case of a quadratic imaginary number field one can construct its maximal abelian extension using torsion points of an elliptic curve with complex multiplication by this field.
In the case of a ...