Questions tagged [cyclotomic-fields]
A cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to $\mathbb Q$, the field of rational numbers.
106 questions
0 votes
0 answers
153 views
Identity involving primitive roots of unity
Let $m,d\in\mathbb N$ with $\gcd(m,d)=1$. Write $\xi_m=e^{2\pi i /m}$, a primitive $m$-th root of unity. Let $r_1,r_2$ be integers with order exactly $d$ in $\mathbb Z_m^\times$ (i.e. $r_i^d\equiv 1\...
- 2,589
1 vote
0 answers
256 views
Is Alexander Stolin's proof of the Kummer–Vandiver conjecture valid? [closed]
In https://arxiv.org/abs/2001.09702 Alexander Stolin announced a proof of the Kummer–Vandiver conjecture. My questions are: Is his proof valid? And what is the status of the Kummer–Vandiver conjecture?...
- 199
13 votes
0 answers
520 views
On mechanically checking Fermat's last theorem for large exponents
I first note that I will solely focus on the second case of FLT, as the concept of Wieferich prime pretty much satisfies my curiosity in the first case and suggests what kind of answers I am looking ...
- 480
5 votes
1 answer
333 views
Unramified extension of $\mathbb Q(\mu_{37})$ of degree $37$
It is well known that $37$ is the first irregular prime, i.e. a prime number $p$ which divides the class number $h_K$ of $K = \mathbb Q(\zeta_{p})$. For $p=37$, the Hilbert class field $L$ of $K$ is ...
- 453
5 votes
1 answer
406 views
Numerical methods for nearest cyclotomic integer
Suppose that you have the decimal expansion of a complex number $z$ to arbitrary precision. Are there any known numerical methods for computing a good or even best approximation of $z$ by a cyclotomic ...
1 vote
0 answers
135 views
Order 5 cyclotomic unit and the Rogers-Ramanujan identity
I asked this on math.SE two weeks ago, with no feedback except 43 views and an upvote. I would understand if this will be closed here as too vague and/or shallow. Still, here it is. The question is ...
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28 votes
3 answers
2k views
A Conjecture Involving Odd Primes Ending in $1$
Context and Motivation Consider the function: $$ f(n)= (a^n + a^{-n})(b^n + b^{-n}) ,$$ where, \begin{align*} a & =\tan{9^\circ}=\tan{\pi/20}=1+\sqrt{5}-\sqrt{5+2\sqrt{5}}\\ b & =\tan{27^\circ}...
- 559
9 votes
0 answers
300 views
Fixed non-trivial ideal classes of $\mathbf{Z}[\zeta_p]$ over $\mathbf{Z}$ under the action of the Galois group
Can a non-trivial ideal class (i.e., an ideal class different from the class of principal ideals) of $\mathbf{Z}[\zeta_p]$ where $p$ is prime, be fixed under the action of the Galois group (when ...
2 votes
0 answers
124 views
Is there a pure algebraic proof of the problem about class number of imaginary quadratic field $p\nmid h_{\mathbb{Q}(\sqrt{-p})}?$
For $p\equiv3\pmod4$, let $h_{\mathbb{Q}(\sqrt{-p})}$ denote the class number of $\mathbb{Q}(\sqrt{-p})$, the class number formula shows $h_{\mathbb{Q}(\sqrt{-p})} < p$. I wonder if there is a pure ...
- 21
0 votes
0 answers
97 views
Cyclotomic eigenvalue question for Distance-regular graph
I have read this paper. So, I am just thinking about if the following guess is true: GUESS: Any Distance-regular graph (DRG) has cyclotomic character value property (which means the eigenvalues of a ...
- 109
6 votes
2 answers
545 views
About the solutions of $ \dfrac{x^p - y^p}{x - y} = a^2+pb^2 $
I already posted this question on MSE. Using theorem $IV$ from this article, it is possible to prove that when $p$ is a prime such that $p ≡ 3\bmod4$, $x ≢ y\bmod{p}$ and $\gcd(x,y) = 1$, then the ...
- 557
3 votes
0 answers
143 views
The maximal $p$-abelian $p$-ramified extension of the cyclotomic $\mathbb{Z}_p$-extension
$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\Cl{Cl}$[Reference: S. Lang, Cyclotomic Fields I and II, §2 chap 6] Let $K$ be a number field. Suppose that $K$ contains the $p$-th roots of unity if $...
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1 vote
0 answers
71 views
Finding the radical expressions of trig functions [closed]
I am trying to find the exact radicals of the sine and cosine of (m/n)*pi. I have been using sympy to do this and made this pretty good script: ...
6 votes
0 answers
296 views
Abelian extensions of Q and cyclotomic fields
I have changed some notation based on the comments of Chris Wuthrich and Wojowu. For an abelian extension $F$ of $\mathbb{Q}$, let $c(F)$ be its conductor. That is, $c(F)$ is the smallest positive ...
- 395
2 votes
0 answers
137 views
Abelian extensions of the rationals
Let $E$ and $F$ be finite abelian extensions of $\mathbb{Q}$ such that $E\cap F=\mathbb{Q}$. (You could take $E$ and $F$ as cyclotomic fields if that makes my question easier.) Set $K:=EF$ (the field ...
- 395
5 votes
0 answers
479 views
Meaning of a result of Gauss on "Mensura" of cyclotomic numbers
(This question was asked before on Math StackExchange. I received a few useful comments there, which helped me answer it for a special case, but I did not succeed in proving the general case.) In an ...
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5 votes
1 answer
437 views
Diophantine equation $\cos(2\pi x)\cos(2\pi y) = \cos(2\pi z)$
While working on finite order elements of $\operatorname{SO}_n$, I meet this question: Find all identities of the form $\cos(2\pi x)\cos(2\pi y) = \cos(2\pi z)$ with $x, y, z$ rational numbers. As ...
- 3,472
6 votes
0 answers
219 views
Is the minus class group isomorphic to the relative class group?
I think this is something I should have known, but if I ever did I forgot about it. Consider the field $L$ of $p$-th roots of unity ($p$ prime) and its maximal real subfield $L^+$. The transfer of ...
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0 votes
1 answer
188 views
Existence of linear disjointness between an algebraic number field and $p$-cyclotomic field over $ \mathbb{Q}$
Suppose $ K $ be an algebraic number field and $ n $ be an even integer. Is it possible to find atleast one $ p $ such that $ p\equiv 1( \text{mod}~ n)$ and $ \mathbb{Q}(\eta_p) $ is linearly disjoint ...
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4 votes
0 answers
120 views
How to describe a concrete generator of $\widetilde{K_0(\mathbb{Z}[C_{23}])} \cong \widetilde{K_0(\mathbb{Z}[\zeta_{23}])}$
Milnor (page 29, see below) gives an explicit proof that the zeroth $K$-theory of the group ring $\mathbb{Z}[C_p]$, where $C_p$ is the cyclic group of order $p$ with $p$ a prime agrees with $K_0$ of $\...
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0 votes
1 answer
224 views
Not containing $ 2^{i}$-th primitive roots of unity in cyclic galois extension of number field
Suppose $K$ is an algebraic number field. We want to determine whether, for all even $n$, there always exists a cyclic Galois extension $L$ of degree $n$ over $K$ such that the intermediate field $K_2$...
- 923
4 votes
0 answers
97 views
Units in Abelian extensions which are not in the subgroup of cyclotomic units
This question is motivated by a Quora post and the top answer to it. The post wishes to make explicit the difference between units in cyclotomic fields and cyclotomic units. One problem with answering ...
- 1,686
5 votes
1 answer
722 views
Class numbers of cyclotomic fields and their maximal totally real subfields
Let $\zeta_p$ be a $p$-th root of unity for a prime $p$, let $L:=\mathbb{Q}(\zeta_p)$ and $K$ the maximal totally real subfield of $L$, i.e. $K:=\mathbb{Q}(\zeta_p+\zeta_p^{-1})$. I am trying to prove ...
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3 votes
0 answers
220 views
relating class number and narrow class number of a real field
I am interested in finding out when the narrow class number of $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ is the same as the class number of $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ where $\zeta_p$ is a primitive $...
- 667
3 votes
0 answers
270 views
Generalisation of Sharifi's conjecture for Siegel varieties
I recently learned Sharifi's conjecture in this article by F.Takako and K.Kato. According to my understanding, this conjecture states that (the conjugation invariant) of the projective limit $\...
- 1,310
0 votes
0 answers
119 views
Relation between two finite abelian extensions of rationals
For $\mathbb{F}$ a finite abelian extension of $\mathbb{Q}$, recall that the conductor $c(\mathbb{F})$ of $\mathbb{F}$ is the smallest positive integer such that $\mathbb{F}$ is contained in the $c(\...
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3 votes
0 answers
184 views
Recover cyclotomic integer with bounded coefficients from its known associate
Recall that two cyclotomic integers are called associated if their ratio is a unit in the corresponding ring of cyclotomic integers. We will view cyclotomic integers as polynomials (of degree $<\...
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2 votes
0 answers
181 views
Narrow class number of a the maximal totally real number field inside a cyclotomic field
I am wondering how much it is known about the narrow class number of the number field $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ ($p$ an odd prime). More precisely, I am interested to know when it is odd. By ...
- 667
5 votes
1 answer
1k views
The group of all units of integral cyclotomic ring
Let $\zeta_n = e^{i2\pi/n}$. What is the group of all units in the integral cyclotomic ring $\mathbb{Z}[\zeta_n]$? Here I like to know all the group elements for small $n$'s. For $n=1$ and $n=2$, the ...
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0 votes
0 answers
138 views
frobenius map on primitive nth root of unity over Fp(w) with (n,p)=1
Suppose w is a primitive n-th root of unity, let r = Ord_n(p) (the smallest integer s.t. $p^r=1 \pmod n $), $f(w)=w^p$(the ...
- 1
3 votes
0 answers
142 views
Minimal Norm Vectors in certain Cyclotomic Ideal Lattices
Let $q$ be an odd prime and let $\zeta$ be a primitive $q^{th}$ root of unity. Let $I$ be the ideal in $\mathbb{Z}[\zeta]$ generated by $1-\zeta$. It is known that we can give $I$ the structure of an ...
7 votes
0 answers
186 views
Finding when a certain product in a cyclotomic field is equal to one
For the following, fix $m\geq 2$, and let $a_{0},\dots,a_{m-1}\in\mathbb{Z}$ be such that $\sum_{k=0}^{m-1}a_{k}=0$. I would like to find the exact conditions on the $a_{k}$ so that the following ...
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4 votes
2 answers
662 views
Existence of solution for a system of quadratic diophantine equations / symmetric quadratic froms
I am interested in solving, or even just deciding the existence of a solution, for a system of quadratic diophantine equations. Let $p$ be a prime congruent to 1 modulo 8, so $ p =17$ is the first ...
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1 vote
0 answers
110 views
How to describe the automorphisms of $\mathbb{Q}(\theta)$ that fix $\mathbb{Q}(\sqrt{-m})$
Suppose $m > 0$ is a square free integer and $m \equiv 1 $mod $4$. Then the $K = \mathbb{Q}(\sqrt{-m})$ is a subfield of $L = \mathbb{Q}(\theta)$, where $\theta$ is a primitive $4m$-th root of ...
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2 votes
0 answers
109 views
Question about infinitude of $m$-irregular primes
Let $p>2$ be a prime, $\zeta$ a primitive $p$th root of unity, and $m >0$ a square-free integer such that $m \not\equiv 3 \mod 4$ and $\gcd(m,p)=1$. Let $\chi$ be the imaginary quadratic ...
- 717
0 votes
1 answer
159 views
The average number of solutions to $a+b=c$ in a multiplicative subgroup of $\mathbb{F}_q^\times$ when $c\in\mathbb{F}_q^\times$ is random
$\mathbb{F}_q^\times$ is the multiplicative group of the finite field $\mathbb{F}_q$, and H is a multiplicative subgroup of $\mathbb{F}_q^\times$ of order $r<q−1$. What is the average number of ...
1 vote
0 answers
118 views
"multi-dimensional" cyclotomic number
Let $F_q$ be the finite field with $q$ elements with characteristic $p$ and with $g$ being a primitive root. Let $N$ be a divisor of $q-1$ and let $C_0$ be the subgroup of $F_q^*$ with index $N$. Then ...
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1 vote
0 answers
280 views
bound norm of algebraic integers in cyclotomic field
Let $\zeta$ be the $p$th root of unity, with $p$ an odd prime number. Let $\mathbb{Q}(\zeta)$ be the $p$th cyclotomic field and let $\mathcal{O}=\mathbb{Z}(\zeta)$ the ring of integers of $\mathbb{Q}(\...
- 19
1 vote
1 answer
461 views
Classification of cyclotomic fields with class number 1
1.Let $p$ be an odd prime number. Is there a classification of cyclotomic fields of the form $\mathbb{Q}(\zeta_p)$ with class number 1? 2.Is there such a classification for general cyclotomic fields $...
- 331
2 votes
1 answer
435 views
How can I prove this claim about splitting of prime ideals in real cyclotomic fields?
Let $L_k = \mathbb{Q}(\zeta_{2^k} + \zeta_{2^k}^{-1})$ be the maximal real subfield of the cyclotomic field of conductor $2^k, k \ge 2$ and $f_k(x)$ be the minimal polynomial of $\zeta_{2^k} + \zeta_{...
user106850
6 votes
0 answers
569 views
Galois cohomology with coefficients in the unit group of a cyclotomic field
While understanding Fermat's last theorem's proof for regular primes, I bumped into a proposition (http://www2.biglobe.ne.jp/~optimist/algebra/fermat2_proof.html#proof10, written in Japanese) stating: ...
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3 votes
1 answer
333 views
Connecting different ways of constructing cubic extensions of $\mathbb{Q}$
There are at least two ways to construct cyclic cubic extensions of $\mathbb{Q}$ as explained below. (A third one is given in the answer to an earlier question). Given $A, B, C$ integers with $A\neq ...
- 1,686
24 votes
1 answer
941 views
Why is the regulator of the degree 11 extension of $\mathbb{Q}$ so large?
Let $K_\infty/\mathbb{Q}$ denote the $\hat{\mathbb{Z}}$-extension of $\mathbb{Q}$. Then for each $n\geq1$, $K_\infty$ has a unique subfield $K_n$ of degree $n$ over $\mathbb{Q}$. The fields $K_n$ are ...
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10 votes
3 answers
935 views
Realizability of a real representation using real cyclotomic coefficients
Let $G$ be a finite group and $\rho: G \rightarrow GL(d,\mathbb{C})$ an irreducible representation with Frobenius-Schur indicator $\frac{1}{|G|}\sum_{g\in G} \operatorname{tr} \rho(g^2) = 1$. Thus $\...
- 145
4 votes
1 answer
243 views
Multiplicative set of positive algebraic integers
Let $S$ be a set of algebraic integers such that: $\mathbb{N}_{\ge 1} \subseteq S \subset \mathbb{R}_{\ge 1}$, $\alpha, \beta \in S \Rightarrow \alpha \beta \in S$, $\alpha, \beta \in S \Rightarrow ...
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3 votes
0 answers
169 views
What are all the possible indices for the finite depth subfactors?
Jones index theorem (1983) states that the set of all possible (finite) indices of subfactors is exactly $$\mathrm{Ind}=\{ 4 \cos(\pi/n)^2 \ | \ n \ge 3 \} \cup [4, \infty),$$ but if we restrict to ...
- 27.6k
9 votes
1 answer
285 views
Are the class numbers of $\mathbb{Q}(\cos(2\pi / m))$ $O(m^n)$ for some fixed $n$?
Question: Are the class numbers of $\mathbb{Q}(\cos(\frac{2\pi}m))$ $O(m^n)$ for some fixed $n$? Evidences (e.g. a recent paper) showing that the question above is open are also OK. Remark: If such $n$...
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13 votes
0 answers
767 views
Is class group of cyclotomic fields cyclic?
What are the cyclotomics fields with a cyclic class group. I read that there are only 29 cyclotomic extensions of $\Bbb Q$ with class number one. But I wanted to know what condition on $n$ would make $...
- 763
0 votes
0 answers
162 views
Elliptic units as Euler systems
I’m trying to understand elliptic units in order to work with the Euler systems of the abelian extensions of quadratic imaginary number fields. I’ve looked at few references about the topic, but they ...
- 99
1 vote
0 answers
225 views
Vandermonde shift
I'm looking for any known results on a shift operator commutated by a Vandermonde matrix. That is, let $$T=\begin{bmatrix}0 & 1 & 0 & 0 & \cdots \\ 0 & 0 & 1 & 0 & \...
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