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A cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to $\mathbb Q$, the field of rational numbers.

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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\...
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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?...
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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 ...
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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 ...
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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 ...
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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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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}...
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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 ...
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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 ...
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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 ...
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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 ...
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$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\Cl{Cl}$[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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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: ...
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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 ...
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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 ...
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(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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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 ...
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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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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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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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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 $K_2$...
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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 ...
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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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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 $...
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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 $\...
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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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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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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 ...
did's user avatar
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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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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 ...
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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 ...
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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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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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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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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 ...
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$\mathbb{F}_q^\times$ 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 ...
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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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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}(\...
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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 $...
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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_{...
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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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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 by Margerie Mumblecrust to an earlier question.) Given ...
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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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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 $\...
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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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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 ...
Sebastien Palcoux's user avatar
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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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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 $...
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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 ...
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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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