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### Remark 4.23.4 in Hartshorne

Crosspost from math.stackexchange, since it's quite possible I might not get a response there.
Remark 4.23.4 in Chapter IV of Hartshorne's Algebraic Geometry references a paper by Elkies that ...

**7**

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**2**answers

285 views

### Field of definition of dominant morphisms

Let $k$ be an algebraically closed field and $k_0$ a sub-field. Let $X,Y$ be two projective varieties defined over $k_0$. Suppose that that there exists a dominant morphism $f$ between $X_k=X\otimes ...

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424 views

### Intuition behind Kronecker's congruence?

The modular polynomial is defined by$$\Phi_n(X, \tau) = \prod_{\tau} (X - j(\tau)),$$where $j$ is the elliptic modular function and $\tau$ is running through classes of imaginary quadratic integers of ...

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votes

**1**answer

284 views

### Why there are only finitely many $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves with CM by $\mathcal{O}$?

For someone who does not have a very extensive knowledge of number theory, what is a good intuitive explanation as to why there are only finitely many $\overline{\mathbb{Q}}$ isomorphism classes of ...

**1**

vote

**1**answer

86 views

### Freeness of the group of principal ideals of a number field

This is just a question wondering whether a concrete (enough) result that can be proved using the axiom of choice can be proved without it. The result being that the group of principal ideals $P_K$ of ...

**0**

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**1**answer

87 views

### Computation of Hilbert symbol of order 4

We have explicit expressions for the quadratic Hilbert symbol over $\mathbb Q$, for example $\left(\dfrac{x,y}2\right)_2=(-1)^{\frac{x-1}2\frac{y-1}2} (x,y\ne2)$. Are similar expressions known for ...

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237 views

### Result of Deuring, intuitive way to see it's true/quickest way to prove?

There is the following result of Deuring that goes as follows:
Let $E/L$ be an elliptic curve defined over a number field $L$ with complex multiplication by an order $\mathcal{O}$ in an imaginary ...

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votes

**1**answer

95 views

### How to check if argument of complex number is rational multiple of $\pi$ [closed]

Assume there is give a complex number $z\in \mathbb{C}$ of modul 1. Therefore it can written as $e^{i2\pi x}$. What are the known criteria for deciding if $x$ is a rational number???

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188 views

### Genus of $k(T)$ is $0$ without using Riemann-Roch

Let $F$ be a function field with one variable with total constant field $k$, and let $X$ be the set of all places of $F$. How do I show that the genus of $k(T)$ is $0$ without using Riemann-Roch? Is ...

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399 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 $(\mathbb{Z}/4\mathbb{Z})^\times \to \mathbb{C}^\times$ ...

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**1**answer

85 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 ...

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**0**answers

93 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 ...

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59 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 ...

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**1**answer

160 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 ...

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**1**answer

447 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 ...

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43 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 ...

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110 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 ...

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80 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 $K$, ...

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122 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 $K$ ...

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73 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 ...

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**1**answer

255 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 ...

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**2**answers

250 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 ...

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38 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 ...

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**0**answers

67 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 $K/k$, ...

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227 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 ...

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### 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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### 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 ...

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**1**answer

116 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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128 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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87 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 ...

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152 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. ...

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**1**answer

244 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 ...

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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 ...

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613 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 ...

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**1**answer

193 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 ...

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**1**answer

454 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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226 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 ...

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**1**answer

477 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 ...

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**1**answer

124 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 ...

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195 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?

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### 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 ...

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517 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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108 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 ...

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884 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 ...

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**1**answer

165 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 ...

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**1**answer

352 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 ...

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**1**answer

335 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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107 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 ...

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188 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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417 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,
...