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.
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Under which circumstances can the class number of a quadratic form not degenerate into the class number of imaginary quadratic fields?
The number of primitive reduced forms with discriminant $D< 0$ is called the class number of discriminant $D$. We denote it by $h_f(D)$.
Write down a Delta symbol
$$
\Delta(D)=\begin{cases}
\...
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Characters of $\mathbb{Q}^{n}$
Just as the characters of $\mathbb{Q}$ are in one-to-one correspondence with the adeles $\mathbb{A}_{\mathbb{Q}}$, which is the restricted product $\mathbb{R} \times \prod_{p\in\mathbb{P}}\mathbb{Q}_{...
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Is this example of 1-dimensional geometric class field theory correct?
I'm trying to understand the main theorem of geometric class field theory. Could someone tell me if this example is correct?
Main theorem. Let $K$ be a function field of a curve over a finite field. ...
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The two conductors are one
It's an exercise (N. Childress Ex. 6-8 p. 152)
I have to prove that two notions of conductor are in fact the same one.
$\let\ss\subset \let\f\frac\let\r\sqrt\let\w\wedge\let\imp\Longrightarrow\let\fa\...
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What is so special about $\mathbb{Q}(\sqrt{398})$ that it has several good prime generating polynomials?
Going through an old post of mine from 2014, I realized there was a curiosity that hasn't been fully explained up to now. Consider the following simple prime-generating polynomials. (My thanks to ...
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Prime-generating polynomials like $F(n) = 7n^2+49n+41$ of real quadratic fields with class number 1?
I just answered an old post Relatives of Heegner numbers? about prime-generating polynomials of form,
$$F(n)=n^2+n-p^2$$
for $p = (2,3,5,7,13)$. It turns out that since the discriminant of $F(n)$ is $...
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The number of imaginary quadratic fields of class number 3 is finite ??
I read here
The primes $p$ of the form $p = -(4a^3 + 27b^2)$
that
" It is known that the number of imaginary quadratic fields of class number 3 is finite. "
But the links did not show it.
...
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Algebraically closed field [closed]
Is there an example of a field $E$ which is not algebraically closed but each finite simple field extension $K$ of $E$, is isomorphic (not necessarily is identity over $E$) as a field to $E$?
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Integration over restricted direct products is only useful for specific functions
So I've been reading Tate's thesis currently. In that we have defined integration of functions on $G$, which are basically formed from restricted direct products of locally compact groups $G_{\...
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Query in Kronecker-Weber Theorem
I am reading Culler's proof of the Kronecker-Weber Theorem here:(https://www.math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALFULL/Culler.pdf) and in the last statement for the Kronecker-Weber ...
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Understanding a lemma in Sharifi's Class Field Theory notes
Here's the lemma: Let $G$ be a finite group, $g\in G$ with image $\bar{g}\in G^{\text{ab}}$, and let $\chi:G\to\mathbb Q/\mathbb Z$ a homomorphism. Viewing $\bar{g}$ as an element of $H_T^{-2}(G,\...
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Corollary 8.7 Cox [duplicate]
Corollary 8.7 says the following:
Let $L,M$ abelian extensions of $K$. Then $L \subseteq M$ $\iff$ there exists a modulus $\mathfrak{m}$ divisible by all primes of $K$ ramified in either $L$ or $M$, ...
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Explicit calculation of norm groups in Q_p
Let $p$ be a prime and $(p, n)=1$. I am wondering how to explicitly compute the norm groups of $\mathbb{Q}_p[\zeta_n]/\mathbb{Q}_p$. Ideally, I would like a computation using class field theory and ...
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What do the products in a modulus in Class Field Theory mean?
I am reading Cox's Primes of the form $x^2+ny^2$. There, he's given the following definition:
Given a number field $K$, a modulus in $K$ is a formal product $$\mathfrak{m}=\prod_{\mathfrak{p}}\...
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Clarification regarding statement in class field theory
Let $K$ be an imaginary quadratic field, $\mathcal{O}_K$ be its ring of integers, $\mathcal{O}$ be an order, $I_K, P_K$ be the group of ideals and principal ideals in $\mathcal{O}_K$, $I_K(m)$ be the $...
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Infinite primes in number fields
David Cox's Primes of the form $x^2+ny^2$ defines infinite primes as
[Infinite primes] are determined by the embeddings of $K$ into $\mathbb{C}$. A real infinite prime is an embedding $\sigma: K \to \...
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Compositum of field extensions in context of $\mathbb Z_p$ extension
Suppose I have a $\Gamma \simeq \mathbb Z_p $ extension $F_\infty /F$ of a number field $F$. Let $F_n$ be the fixed field of $\Gamma _n \simeq p^n \mathbb Z_p$. Denote by $L_n$ the maximal unramified ...
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Bijection between Binary Quadratic Forms and narrow Class group of Quadratic Number Fields over a non-$\mathbb Q$ base.
It is well known that there is a bijection between $GL_2(\mathbb Z)$-equivalence classes of binary quadratic forms with $\mathbb Z$ coefficients of a fixed discriminant $D$, and the narrow class group ...
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Showing the first cohomology group of roots of unity in a cyclotomic $\mathbb Z_p$ extension is trivial
Let $F_\infty$ be the cyclotomic $\mathbb Z_p$-extension of $F$ - a number field such that $\mu _p \subset F$. Let $W$ be the set of all roots of unity in $F_\infty$.
I want to prove that $H^1 (\...
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The number of extensions of $\mathbb{Q}$ with degree $n$ and discriminant dividing $D$
In this question, I asked about methods to classify finite abelian extensions of $\mathbb{Q}$ (you don't really need to read the question though) and Offlaw provides a nice answer and claim that
The ...
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Classifying abelian extensions of number fields with class field theory
I am reading about local class field theory here https://math.mit.edu/classes/18.785/2015fa/LectureNotes24.pdf and in the end the author mentions an application to count the number of Galois ...
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There are infinitely many prime number $p$ such that $p≡3\pmod4$ and $p$ splits completely in $K$.
Let $K$ be a number field which does not contain $\Bbb{Q}(i)$.
Then, I want to prove there are infinitely many prime number $p$ such that $p≡3\pmod4$ and $p$ splits completely in $K$.
To prove this ...
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Norm residue map maps uniformizing elements to Frobenius elements
I am reading Algebraic Number Theory by Cassel, Frohlich. I have a question about the proof that norm residue map maps uniformizing elements to Frobenius elements in unramified extension. This is in ...
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Is there an algebraic function that satisfies the Abelian complex multiplication with even irrational multiplicator?
According to Abel's astonishing paper "RECHERCHES SUR LES FONCTIONS ELLIPTIQUES.[1]", we can get
\begin{align*}
\varphi_3(x,\kappa_3)&=\frac{x\cdot\left(\sqrt{3}-\,\kappa_3x^{2}\right)}{...
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Cohomology of local fields in positive characteristic
It is well-known from local class field theory that the Brauer group $\text{Br}(k)$ of a local field $k$ gets killed as you pass to sufficiently large extensions of $k$. In particular, $\text{Br}(L)(p)...
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Tate cohomology of units
Show that for any local field K, there is a finite Galois extension L, such that $H^{i}_{T}(Gal(L/K),O_L^*)$ does not vanish for all i, here $O_L$ is ring of algebraic integers of L.
I only know that ...
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uniqueness of local Artin map
This problem defines a map with the same properties as local Artin map and asks you to prove they are equal. I'm having problems with b) and c). Is the first part of b) comes from Galois ...
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A field has a unique algebraic extension of each degree if and only if its absolute Galois group is the profinite completion of integers
I would like to know how to prove that a field $K$ which has a unique algebraic extension of each degree has its absolute Galois group isomorphic to the profinite completion of the integers $\hat{\...
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Basic proofs on Weil groups
I had some questions regarding the Weil group. We defined a surjective map $res: \operatorname{Gal}(L/K) \rightarrow \operatorname{Gal}(k_L/k)$, where $k_L,k$ is the residue field of $L,K$ ...
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Find a non-trivial element in the class group of $\mathbb{Q}( \sqrt{−5})$.
a. Find a non-trivial element in the class group of $\mathbb{Q}(\sqrt{−5})$.
b. Show that the class group of $\mathbb{Q}( \sqrt{−5})$ has order two.
For part a: I know that the class group is the ...
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Class group of quadratic field extension $\mathbb Q(\sqrt{-69})$ has order 8
I'm practicing finding class groups, in this case for $K = \mathbb{Q}(\sqrt{-69})$, and found the class number $h_K$ to be 16, with $C_K \cong C_2 \times C_2 \times C_4$ (cyclic product) whereas https:...
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Why $Br(Spec(K))=Br(K)$ holds?
This question may be very elementary, sorry but I would like to ask this question.
Brauer group of local ring $(R,m)$ is defined to be a group of equivalent classes of Azumaya algebra over $R$. Local ...
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Step in Yoshida's proof of Hasse-Arf theorem
This question concerns Yoshida's proof of the Hasse-Arf theorem in the local class field theory in https://arxiv.org/abs/math/0606108 (page 16).
For a totally ramified extension $K′/K$ of local fields,...
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More class field theory
The classical language of class field theory can be generalized by class formation. So far I've seen, for example, the absolute Galois group acting on the multiplicative group of a local field, or the ...
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Is there an reciprocity in topology?
Let $K$ be a global field. As explained in this nLab page, Artin reciprocity is an isomorphism between two abelian groups
$$K^\times \backslash \mathbb{I}_K / \mathcal{O} \xrightarrow{\sim} \...
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Why Chebotarev's density theorem implies $ \exists {v \in Ω_L}$ such that $Gal(L_v/K_{v'})$ is generated by $\sigma^2$?
Let $L/K$ be an extension of number field.
Let $Gal(L/K)$ is generated by $\sigma$.
Why Chebotarev's density theorem implies $ \exists {v \in Ω_L}$ such
that $Gal(L_v/K_{v'})$ is generated by $\sigma^...
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What does the characteristic ideal of a f.g. torsion $\Lambda(G)$-module tell me about the arithmetic of the extension?
I am currently trying to learn Iwasawa theory and am following J. Coates and R. Sujatha's book 'Cyclotomic Fields and Zeta Values'.
The setup is the following:
Let $\mathcal{F}_n:=\mathbb{Q}(\mu_{p^{...
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Notation in the Tate-Nakayama Theorem
In ch. IX, §8 of Serre’s Local Fields, we find the Tate–Nakayama theorem, an essential lemma for class field theory:
Theorem 14. Let $G$ be a finite group, $A$ a $G$-module, and $a\in H^2(G,A)$. Let $...
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How does the approximation theorem of valuations ensure both a lower bound and an upper bound?
I am working my way through Artin & Whaple's 1945 paper [1] that characterizes global fields based on axioms pertaining to the set of places on a given field $K$. But I'm having difficulty ...
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Chebotarev's Density Theorem, Equidistribution of Prime Ideals, and Class-Field Theory
I am working on a senior thesis, and my advisor told me to look into the theory that prime ideals in a number field of norm less than $N$ are evenly distributed across ideal classes.
I've looked at ...
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Inflation-restriction sequence for profinite groups
I recently learnt about profinite group cohomology to do class field theory and I am looking for a proof of the profinite version of the inflation-restriction sequence which hopefully just uses the ...
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Splitting of the ring class group of an imaginary quadratic field
Let $K$ be an imaginary quadratic field, $\mathcal{O}_K$ be its ring of integers and $\mathcal{O}(n)$ be an order of $K$ of conductor $n\in\mathbb{N}$. It is known that there is a surjection between ...
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Even more mysterious sum equaling to $\frac{7(p^2-1)}{24} - h(-p)$ where $p \equiv 3 \pmod{4}$
This is a direct follow-up to this question, which I think is interesting enough to be a separate problem. Please check that post out too in case there are ideas there that might transfer here.
In the ...
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Proof of quadratic reciprocity from Artin reciprocity
I just read the proof of the quadratic reciprocity from the Artin reciprocity here Eisenstein and Quadratic Reciprocity as a consequence of Artin Reciprocity, and Composition of Reciprocity Laws given ...
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Question on complex multiplication and the Ray class field of imaginary quadratic fields.
I am beginning to study the theory of complex multiplication, and I haven't yet got my hands on a copy of Silverman's advanced topics. For now I'm making do with chapter 6 in Silverman-Tate, Rational ...
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Corollary 8.7 in Cox's primes of the form $x^2+ny^2$
Corollary 8.7 in Cox's primes of the form $x^2+ny^2$: given a number field $K$ and two abelian extensions $L, M$, $L\subseteq M$ if and only if there is a modulus $\mathfrak m$ divisible by all primes ...
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Number of extensions of a place
Let $L/K$ be a separable extension (not necessarily normal). Let $v \in K$ be a place of $K$ and let $W_{v}$ be the set of places of $L$ extending $v$. In a remark in Algebraic Number Theory by ...
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Relation between Frobenius elements and double cosets
Some days ago I was reading Serre's book on local class field theory and I got stuck on a point he makes in chapter seven after having defined the transfer homormophism.
First I report some notation:
...
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Local invariant map in the case of closed points of a curve
In class field theory, we have the well-known local invariant map $\mathrm{inv}_v: \mathrm{Br}(k_v) \rightarrow \mathbb{Q}/\mathbb{Z}$, where $k$ is a field and $v$ is a place of $k$. Similarly, we ...
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Existence of a totally complex cyclic cyclotomic extension
I am reading J. S. Milne's Class Field Theory and have a question about his proof for Lemma 7.3, Chapter VII https://www.jmilne.org/math/CourseNotes/CFT.pdf:
Lemma 7.3: Given a number field $K$, a ...
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