Questions tagged [class-field-theory]

Class field theory is a major branch of algebraic number theory that studies abelian extensions of global and local fields.

Class field theory is a major branch of algebraic number theory that studies abelian extensions of global fields and local fields. It also includes a reciprocity homomorphism which acts from the idele class group of a global field to the Galois group of the maximal abelian extension of the global field.

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Serge Lang Never Explains Anything Round II

I'm reading the second edition of Lang, Algebraic Number Theory, page 221. I quote: Let $F$ be a local field, i.e. the completion of a number field at an absolute value. Let $L$ be an abelian extension with Galois group $G$. Then there exists a…
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Which number fields can appear as subfields of a finite-dimensional division algebra over Q with center Q?

I have some idle questions about what's known about finite-dimensional division algebras over $\mathbb{Q}$ (thought of as "noncommutative number fields"). To keep the discussion focused, let's concentrate on these: Which number fields $K$ occur as…
Qiaochu Yuan
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Class field theory for $p$-groups. (IV.6, exercise 3 from Neukirch's ANT.)

I will use notation as in a preious question of mine. This question is from Neukirch's book "Algebraic number theory," page 305, exercise 3. Notation for the problem Let $G$ be a profinite $p$-group (so all quotients by open subgroups are…
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An interesting table of Prime Generating polynomials similar to $n^2+n+41$?

Here is some data on quadratic prime-generating polynomials of a particular form. Kindly look at the questions given below it. Note: The discriminant $d$ is square-free and its class number $h(d)$ is also given. $$\begin{array}{|c|c|c|c|}…
Tito Piezas III
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Relationship between intersection and compositum of fields

This issue came up in a number theory lecture today. Let $K$ be a number field and let $L/K$ be an abelian (finite Galois) extension. Then there exists a primitive $m$th root of unity $\zeta_m$ so that $K(\zeta_m)\cap L=K$ so that $m$ satisfies a…
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Galois Group of the Hilbert Class Field

Let $K/\mathbb{Q}$ be a number field with Galois group G and let $L/K$ be the Hilbert class field of $K.$ It is easy to show that $L$ is Galois over $\mathbb{Q}$ and I am interested in knowing this Galois group. We know that the class group of $K$…
Lalit Jain
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What do we know about the class group of cyclotomic fields over $\mathbb{Q}$?

Motivated by this question, I am curious how one can characterize primes that splits completely in the Hilbert class field of $\mathbb{Q}(\zeta_q)$, where $q$ is a prime. Then I realize how much I don't know about this extension over $\mathbb{Q}$! I…
user27126
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How does a Class group measure the failure of Unique factorization?

I have been stuck with a severe problem from last few days. I have developed some intuition for my-self in understanding the class group, but I lost the track of it in my brain. So I am now facing a hell. The Class group is given by $\rm{Cl}(F)=$…
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CFT via Brauer groups vs via ideles

I am interested in the relationship between the following two versions of CFT: Version 1: (Brauer Group Version) Let $K$ be a number field. One constructs, for every finite place $v$ of $K$, a map $inv_v:Br(K_v)\rightarrow \mathbb{Q}/\mathbb{Z}$ in…
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How does topology enter Number theory and how we can grasp its essence?

In infinite Galois theory,main theorem failed and we get a "Krull topology" to mend the main theorem, we even generalized that to make definition for profinite group. In local class field theory, the multiplicative group modulo the norm group is…
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$\mathbb{Q}(i)$ has no unramified extensions

It is a classical result that every extension of $\mathbb{Q}$ is ramified. Put differently: there are no unramified extensions of $\mathbb{Q}$. The classical proof follows from the following two statements: (a) The only number field having…
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Given $d \equiv 5 \pmod {10}$, prove $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ never has unique factorization

With the exception of $d = 5$, which gives $\mathbb{Z}[\phi]$, of course (as was explained to me in another question). I'm not concerned about $d$ negative here, though that might provide a clue I have overlooked. What I thought about was the fact…
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primes represented integrally by a homogeneous cubic form

Expired by this question Show determinant of matrix is non-zero I am moved to ask: Given integers $a,b,c,$ and cubic form $$ f(a,b,c) = a^3+2 b^3-6 a b c+4 c^3 = \left|\begin{bmatrix} a & 2c & 2b\\b & a & 2c\\ c & b & a\end{bmatrix}\right|, $$…
Will Jagy
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history and/or motivation for cohomology in class field theory

I am currently learning (local) class field theory via group cohomology with Milne's notes. I have a number of questions about using group cohomology to prove the main statements of class field theory. The first thing: how do you motivate the…
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Class Field Theory Video Lectures

Are there any good video lectures online discussing algebraic number theory, class field theory, or related topics? Something on the level of the book of Neukirch or Cassels and Frohlich would be preferred.
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