# 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.

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### 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 ...
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### 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 ...
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### 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 ...
1 vote
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### 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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### 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 ...
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### 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
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### "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 ...
1 vote
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### 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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### 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 ...
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### 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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### Factoring cyclotomic polynomials over quadratic subfield

The quadratic subfield of $\mathbb{Q}(\zeta_p)$ is given by $\mathbb{Q}(\sqrt{p^*})$, where $p^*$ is the choice of $\pm p$ which is $1$ mod $4$. By some elementary Galois theory, the cyclotomic ...
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In the following, let $\mathbb{Q_1}$ denote the subfield of degree $p$ over $\mathbb{Q}$ in the $p^2$- cyclotomic extension. What is the best known upper bound for the size of its class group, $\text{... 3 votes 0 answers 115 views ### Extended cyclotomic criterion for unitary categorification According to this paper (Corollary 8.54) the Frobenius-Perron dimension (FPdim) of any object$a$of a fusion category over$\mathbb{C}$is a cyclotomic integer. Now, FPdim($a$) is the maximal ... 3 votes 0 answers 135 views ### Real root of the derivative of a prime cyclotomic polynomial Consider the graph of$y=x^n$with$n$odd, and draw a tangent to its negative arc that crosses the graph at$(1,1)$. Equating the slopes gives us$\frac{x^n-1}{x-1}=nx^{n-1}$at the point of tangency.... 9 votes 1 answer 304 views ### Is an integral sum of periodic vectors always a sum of integral periodic vectors? Update: I have found reference to this problem. It is known as "the Rédei-de Bruijn-Schönberg theorem", which is proved in the following papers: N. G. de Bruijn: On the factorization of cyclic ... 1 vote 1 answer 222 views ### How to find a cyclotomic polynomial of degree d that decompose into d irreducible polynomials in$Z_6$? More specifically, I need the degree$d$to be around 1024. I can easily find the cyclotomic polynomial of degree 1024 that satisfies the above requirement in$Z_2$, i.e.,$x^{1024}+1$, which is equal ... 10 votes 1 answer 410 views ### Is there a pattern for the irreducible factors and degrees of a characteristic polynomial? Let$Log(n) = \sum_{i=1}^r \alpha_i \cdot e_i$, where$n = \prod_{i=1}^r p_i^{\alpha_i}$and$p_i$is the$i$-th prime,$\alpha_i \ge 0$,$e_i$is the$i$-th standard basis vector. For example$6 = 2\... 1k views

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### About real abelian number fields

How can I prove this: Let $K$ be a real abelian number field, $K_1$ be the Hilbert Class Field of $K$, and $J=K_1\cap K(\zeta_b)$. If a prime $p$ divided $[J:K]$ but did not divide $[K:\mathbb{Q}]$, ...
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### Splitting of primes in cyclotomic $\mathbb{Z}_p$-extension

Let $p$ be a prime, $K$ be a finite extension of $\mathbb{Q}$ and $K_{\infty}$ be a cyclotomic $\mathbb{Z}_p$-extension of $K$ i.e. Gal$(K_{\infty}/K) \cong \mathbb{Z}_p$, the group of $p$-adic ...
Let $E$ be an elliptic curve over $\mathbb Q$. Let's look at the group of points of this elliptic curve over $\mathbb Q(1^{1/\infty})$ which we get after adding all roots of unity to $\mathbb Q$. It ...