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24
votes
0answers
327 views

Degree 17 number fields ramified only at 2

The number $17$ is the smallest odd number that occurs as the degree of a number field $K/\mathbb{Q}$ for which the only finite prime that ramifies is $2$. The non-existence for $n < 17$ follows ...
5
votes
1answer
152 views

Positive Primes represented by an indefinite binary form, reducing poly degree from 8 to 4

In his lovely answer at Positive primes represented by indefinite binary quadratic form Noam found that a (positive) odd prime $p$ is represented by the indefinite form $x^2 + 13 x y - 9 y^2$ if and ...
2
votes
0answers
103 views

Global Artin reciprocity law from Local class field theory

Let $K=\mathbb F_q((t)), p -$ prime ideal in $K$, $\psi_p$ be the local Artin map$K_p^* \to Gal(K_p^{ab}/K_p)=G_p \subset Gal(K^{ab}/K)$. Then I define global Artin map $\psi_K$as product of $\psi_p$, ...
1
vote
0answers
153 views

Reference for Local class field theory via witt vectors

I would like to find some books or lecture notes on geometric local class field theory via Witt vectors. I can't find any good paper on this subject.All approaches in the books to local class field ...
6
votes
1answer
302 views

Class groups of orders

In Cox's book "Primes of the form $x^2 + ny^2$", he proves that in a quadratic imaginary field $K$, if $\mathcal O$ is an order of conductor $f \in \mathbb Z$, we have that the class group ...
7
votes
1answer
359 views

Hilbert Class Field Galois over Q?

So if we have a Galois extension $K/\mathbb{Q}$, then the Hilbert Class Field $H$ of $K$ is certainly Galois over $\mathbb{Q}$. But is the converse true? I know many examples of nongalois ...
0
votes
0answers
25 views

necessity of divisibility critria for class number of a real quadratic field

Are all the conditions for divisibility of class number of a given real quadratic field by a given number sufficient only? e.g. it is well known class number is divisible by 2 if the discriminant ...
2
votes
0answers
141 views

Ring class fields of orders

The theory of ring class fields corresponding to orders in imaginary quadratic fields is introduced by, say, [1]. But I'm reading an article [2], in which ring class fields corresponding to orders in ...
0
votes
0answers
34 views

sufficiency of the relative discriminant to be a square of an ideal for an unramified quadratic extension

Is it sufficient for a quadratic extension of a cubic number field to have a relative discriminant as a square of an ideal for being unramified extension (excluding primes dividing 2 for the sake of ...
0
votes
0answers
48 views

divisor class group with modulus

Let $C$ be a smooth projective curve over a field $k$ and $S \subset C$ a finite number of points. A modulus is simply a divisor supported on $S$. What is the divisor class group with modulus? I ...
6
votes
0answers
170 views

n-dimensional local fields

Recently, I hear the concept of $n$-dimensional local fields. It is defined inductively as follows. (1) a $0$-dimensional local field is a finite field. (2) an $n$-dimensional local field is a ...
3
votes
1answer
213 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 ...
5
votes
0answers
249 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 ...
1
vote
0answers
74 views

Reference request: modern idelic formulation of class field theory for function fields

I am looking for a modern reference for idelic class field theory for function fields over finite fields including proofs. For example, the book Algebraic Number Theory of Cassels-Frohlich does not ...
4
votes
2answers
323 views

On class numbers $h(-d)$ and the diophantine equation $x^2+dy^2 = 2^{2+h(-d)}$

Given fundamental discriminant $d \equiv -1 \bmod 8$ such that the quadratic imaginary number field $\mathbb{Q}(\sqrt{-d})$ has odd class number $h(-d)$. Is it true that one can always solve the ...
4
votes
1answer
346 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 ...
0
votes
1answer
157 views

Idelic Artin map

I want to study the idelic Artin map for the, say, 7th cyclotomic field explicitly. In other words I want to see how a particular idele can be constructed which gets sent to the identity element in ...
4
votes
2answers
276 views

How does Tate cohomology fit into a derived categories framework?

I've read through one class field theory text after another, but there's something very non-intuitive for me about cohomology that makes it hard for me to understand why Tate cohomology was invented. ...
1
vote
2answers
282 views

2-class group of a quadratic imaginary extension

Let $p\equiv 5 [8]$ be a prime number, and consider $K=\mathbb{Q}(\sqrt{-p})$. I would like to check that the $2$-Sylow subgroup of the class group $C_K$ has order $2$ (I'm pretty sure it's true). ...
1
vote
0answers
223 views

Ray class field and ring class field

Let $K$ be quadratic number field and let $O$ be an order of $K$. A modulus $m$ of $K$ is a formal product $m_0\cdot m_\infty$ of finitely many finite primes $m_0$ and finitely many infinite real ...
3
votes
2answers
250 views

Galois cohomology of the field of Laurent series

Let $k$ a separably closed field. Do we have that $k((t))$ is of cohomological dimension one?
3
votes
1answer
294 views

Fields whose embeddings into the complex numbers are invariant under complex conjugation

Is there a general notion/description of fields $K$ such that the image of any embedding $K \hookrightarrow \mathbb{C}$ is invariant under complex conjugation, thus inducing an involution on $K$ which ...
6
votes
2answers
363 views

the global m-th power reciprocity law and Quartic Reciprocity Law

I'm reading Cox "Primes of the form $x^2+ny^2$". And I read a chapter about the global m-th power reciprocity law. Now I'm not able to prove the quartic and cubic reciprocity laws. Where can i find ...
7
votes
1answer
443 views

Numbers integrally represented by a ternary cubic form

Given integers $a,b,c,$ and cubic form $$ f(a,b,c) = a^3 + b^3 + c^3 + a^2 b - a b^2 + 3 a^2 c - a c^2 + b^2 c - b c^2 - 4 a b c $$ $$ f(a,b,c) = \det \left( \begin{array}{ccc} a & b ...
1
vote
0answers
115 views

Finite extensions of residue fields of Henselian DVRs

Let $K$ be an Henselian discrete valuation field such that its completion is separable over $K$. Let $F$ be its infinite residue field. Is it true that a finite extension of $F$ is a simple extension ...
7
votes
1answer
540 views

How did Takagi prove Kronecker's Jugendtraum for Q(i)?

In Noah Snyder's historical undergraduate thesis on Artin L-Functions, it mentions that Takagi proved Kronecker's Jugendtraum in the case of Q(i) in his doctoral thesis. Since I don't know how to get ...
3
votes
0answers
443 views

Characterizing primes that split completely vs. primes with a given splitting behavior

Given a finite abelian extension of number fields $L/K$, the prime ideals $\mathfrak{p}$ in $O_K$ split into primes $\mathfrak{P}$ in $O_L$. The number of primes $\mathfrak{p}$ splits into is ...
4
votes
0answers
123 views

Reciprocity Map and Cycle Class Map

This might be a very naive question but here it goes. Let X be a smooth variety of dimension d over a p-adic field. We have the n part of the rerciprocity map: $rec/n: SK_1(X)/n \to \pi^{ab}_1(X)/n$ ...
3
votes
1answer
274 views

Abelianized fundamental group of a curve over a finite field

Let $X$ be a smooth, projective, and geometrically connected curve over a finite field $\mathbb{F}_q$ and fix a geometric point $\overline{x} : \text{Spec } \overline{\mathbb{F}_q} \to X$. Then there ...
3
votes
1answer
380 views

Non-cyclotomic abelian extensions

Suppose $L|\mathbb{Q}$ is an abelian extension of number fields. Then, all the roots of unity are certainly contained in the maximal abelian extension $L^{ab}$ of $L$. Why is it obvious that if $L \ne ...
4
votes
0answers
246 views

Abelian cubic extensions of Q[i],

Recently I was considering cubic extensions $K/Q$ that have discriminant negative of a perfect square. Classifying these curves reduces to solving a Diophatine equation of the form $4a^3+27b^2=c^2$ ...
4
votes
2answers
350 views

Intersection of Hilbert class fields of imaginary quadratic fields

In this question Hilbert class field of Quadratic fields it is mentioned that if $d\equiv 1 \mod 4$ then the Hilbert class field of $\mathbb{Q}(\sqrt{-d})$ contains $\mathbb{Q}(i,\sqrt{d})$. Could ...
9
votes
1answer
629 views

The Class Number One Problem for Real Quadratic Fields

An approach to the Gauß class number one problem for imaginary quadratic fields is to determine the integral points on the modular curve $Y_{nonsplit}(n)$ for a suitable $n$. Here follows a quick ...
12
votes
2answers
887 views

Elliptic Curves with CM and Class Field Theory

Let $K$ be an imaginary quadratic field with Hilbert class field $H$, and let $E$ be an elliptic curve defined over $H$ with complex multiplication by the ring of integers $O_K$ of $K$. It is known ...
9
votes
0answers
609 views

What numbers are integrally represented by $4 x^2 + 2 x y + 7 y^2 - z^3$

This is related to my first MO question and Kevin Buzzard's conjecture at Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $ In December 2010 my question appeared in the M.A.A. Monthly, ...
14
votes
5answers
957 views

What is the “ray” in ray class group?

I have never seen any algebraic number theory book discuss the origin of the term "ray class group." Does anyone know where the word "ray" comes from in this context? I always thought it might be a ...
9
votes
0answers
576 views

How does one understand geometric CFT in terms of modularity?

I have recently asked a question in a similar vein: What makes Geometric CFT easier than CFT? but I'm afraid I wasn't quite ripe to ask it yet. I have since consulted with the following sources: ...
5
votes
3answers
858 views

What makes Geometric CFT easier than CFT?

I've been reading: math.stanford.edu/~conrad/249BPage/handouts/geomcft.pdf in an attempt to shed some geometric light on class field theory. The last paragraph there reads: In case the ground field ...
20
votes
1answer
915 views

Can one prove complex multiplication without assuming CFT?

The Kronecker-Weber Theorem, stating that any abelian extension of $\mathbb Q$ is contained in a cyclotomic extension, is a fairly easy consequence of Artin reciprocity in class field theory (one just ...
3
votes
0answers
214 views

Formal non-CM in local fields

An elliptic curve $E$ with complex multiplication by an imaginary quadratic field $F$ has $\ell$-adic Galois representations that essentially encode the class field theory of $F$ - in other words, the ...
11
votes
0answers
351 views

Lubin-Tate vs cohomological approach to local CFT

Local class field theory ("local CFT") can be developed in various ways, among them is a cohomological approach and an explicit approach due to Lubin and Tate (both can be found in Milne's CFT notes ...
6
votes
1answer
369 views

Generalization of Hilbert 94 and capitulation

Let $L/K$ be a finite, cyclic extension of number fields, say with $\mathrm{Gal}(L/K)=G$. In my context $G$ is actually of order $p$, an odd prime number, but let me state my question for every cyclic ...
12
votes
1answer
439 views

Embedding number fields in fields with class number 1

(Apologies if this question isn't quite research-level: a colleague came across it while preparing a non-examinable bonus lecture on class field theory for an undergraduate algebraic number theory ...
3
votes
1answer
338 views

What is the relation of the absolute Galois group and classical profinite groups?

Consider the absolute Galois group $G = \mathrm{Gal}(\overline{\mathbb{Q}}: \mathbb{Q})$ and $G_p = \mathrm{Gal}(\overline{\mathbb{Q}_p}: \mathbb{Q}_p)$. Abelian class field theory gives us for the ...
8
votes
1answer
495 views

class numbers of $\mathbf{Q}(2^{1/n})$

Calculating the class numbers of $\mathbf{Q}(2^{1/n})$ for small $n$ always yields $1$. Is it true for an infinite number of $n$s? Does applying Iwasawa theory to the false Tate curve tower ...
6
votes
1answer
407 views

Parity of class number of pure cubic fields

A pure cubic field is an algebraic number field of the form $K = \mathbb{Q}(\theta)$ with $\theta^3 = m$, $m \neq \pm 1$. What can be said about the parity (odd or even) of the class number of a pure ...
4
votes
2answers
451 views

How do Brauer groups relate to zeta functions?

There are two approaches to class field theory that I was taught. The first, is the theory of $L$-functions, Dirichlet characters and so forth (which I described succintly in the question What are the ...
15
votes
2answers
1k views

Primes of the form $x^2+ny^2$ and congruences.

The answer of following classical problem is surely known, but I can't find a reference For which positive integer $n$ is the set $S_n$ of primes of the form $x^2+n y^2$ ($x$, $y$ integers) ...
12
votes
5answers
2k views

What is the “reason” for modularity results?

The question is a little wishy-washy, but I take my cues from other popular questions that relate to the philosophy behind the mathematics as Why do Groups and Abelian Groups feel so different? . I ...
1
vote
1answer
166 views

Lower bound on the class group of the p-Hilbert class field of an imaginary quadr. field

Let K be an imaginary quadratic field, A(K) its p-class group, and H(K) its p-Hilbert class field. If rk(A(K))=2, a result due to Arrigoni tells us that p^3 divides the order of the class group of ...