Questions tagged [cyclotomic-fields]

Cyclotomic fields are fields where a primitive root of unity is added to the rational numbers. These fields are common in algebraic number theory.

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Kummer's Lemma and $1+\zeta$

In lecture we were told to think about the following: Kummer's Lemma: Let $p$ be an odd prime and let $\zeta := e^{2\pi i / p}$. Every unit of $\mathbb{Z}[\zeta]$ is of the form $r\zeta^g$, where $r$ ...
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rational elliptic curve: torsion subgroup defined over abelian extensions

I'm thinking about a question, that is if E is a rational elliptic curve, K is a cyclotomic field, how can I find the minimal subfield of K such that the torsion subgroup over K can also be defined ...
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Finding exact radical expressions for $\sin\frac{m\pi}{n}$ and $\cos\frac{m\pi}{n}$ [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: ...
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How to compute the different ideal of the cyclotomic field extension $\mathbb{Q}(\zeta_p)/\mathbb{Q}$? [closed]

Let $p$ be a prime number, $K=\mathbb{Q}(\zeta_p)$ be the cyclotomic field extension of $\mathbb{Q}$ by adding a $p$-th root of unity. There is a notation called different ideal, which is defined to ...
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Statement unclear-"Cyclotomic fields and zeta values"

I am currently reading the book Cyclotomic fields and zeta values by J. Coates and Sujatha. There is a statement(Page 16) that is not clear immediately. Given $f$ in $R$, define $h(T)=\prod_{\zeta\in\...
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An efficient algorithm for finding the generators of the unit group of $\mathbb{Z}[\zeta_p] $ (ring of cyclotomic integers, where p is a prime)

The group of units of $\mathbb{Z}[\zeta_p]$ has a finite number of generators. I am aware there are algorithms that can find these generators but they don't seem to be that efficient. I was wondering ...
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Construct primitive element for intermediate field of cyclotomic field extension

I am studying subfields of cyclotomic extension, especially the following situation: Let $\xi$ be a primitive $m$-th root of unity, and $\chi$ be a multiplicative character on $(\mathbb{Z}/m\mathbb{Z})...
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Same field extension, different irreducible polynomial?

I've come across an issue while doing an assigment for a field theory subject. The issue seems pretty basic to me but I can't figure it out it's solution by myself. Let's consider the field extension $...
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Meaning of some results of Gauss on cyclotomic numbers.

In an unpublished fragment of Gauss entitled "on the theory of complex numbers", written by him around 1810, Gauss made early inroads into the theory of cyclotomic numbers, which are ...
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What are the proper subfields of the cyclotomic field extension generated by a primitive fifth root of unity?

I'm currently studying up for an abstract algebra qualifier exam (and final exam). In my review, I've found this problem. I think it's a good practice problem; it touches upon a few good aspects of ...
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Construct precisely the cyclotomic extension of $\mathbf{Q}$

I know very little about Iwasawa theory. According to what I know, the way to define the cyclotomic extension ($\mathbf{Z}_p$-extension) $K_{cyc}$ of a number field $K$ is to notice that the Galois ...
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Determine a basis for $\Bbb F_p[\omega]/\Bbb F_p$.

Fix a prime $p$. Let $\Bbb F_p$ be the finite field with $p$ elements with $\omega$ a primitive $n^{th}$ root of unity. From Galois theory of finite fields, the extension $\Bbb F_p[\omega]/\Bbb F_p$ ...
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Jacobi sum of power residue symbol in Sagemath

I’m computing Jacobi sum of power residue symbol as follows: ...
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For which finite groups $G$ is $M_n(\mathbb{Q}(\zeta))$ a factor of $\mathbb{Q}[G]$?

I have cross-posted this question on Math Overflow here. For a finite group $G$, the rational group ring $\mathbb{Q}[G]$ has a Wedderburn decomposition: $$ \mathbb{Q}[G] \cong \prod_{i=1}^k M_{n_i}(...
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Finding class group of $\Bbb Q(\zeta_{29})$

I wondered the class group $G_{29}$ of the cyclotomotic field $\Bbb Q(\zeta_{29})$ today (29 October) which is the second non-trivial class group of the sequence of fields $\Bbb Q(\zeta_p)$ where$\...
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Find all units in the ring $\Bbb Z[\omega]$ where $\omega$ is the primitive $p^{th}$ root of unity.

Let $p>0$ be a prime and let $\omega$ be a primitive $p^{\text{th}}$ root of unity. I am trying to find all units in the ring $\Bbb Z[\omega]$. Every element of $\Bbb Z[\omega]$ is of the form $z=...
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How to find a factor of $\zeta - 1$ in $\mathbb{Q}(\zeta)$ for $\zeta := e^{2 \pi i / 5}$

Let $\zeta := e^{2 \pi i / 5}$. Given that $\zeta - 1$ is a factor of $\zeta + 4$ in $\mathbb{Z}[\zeta]$, find another factor of $\zeta + 4$. Remark: This is part of exercise 7 in chapter 3 of Stewart'...
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Complex numbers and congruence, article by G.B. Mathews

I am reading "Notes in connexion with Fermat's last theorem", by G. B. Mathews. (accessible per example on Google books) For some integer $k$, he defines $r = e^{\frac{2 \pi i}{k}}$ and: $$ ...
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Lattice Geometry of $\mathbb{Z}[\zeta_5]$

I was trying to plot all the points of $\mathbb{Z}[\zeta_5]$ and see if there is a nice lattice structure. It is easy for the Gaussian Integers: $\mathbb{Z}[\zeta_4] = \mathbb{Z}[i]$, which is a ...
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Can three antiprisms (uniform polyhedra) fit exactly around an edge, leaving no gaps?

Let $\tau=2\pi$ $=360^\circ$. An $n$-gon antiprism has dihedral angles $$\theta_n = \arccos\left(-\frac1{\sqrt3}\tan\frac\tau{4n}\right)$$ (where an $n$-gon meets a triangle) and $$\phi_n = 2\arccos\...
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Are norms of elements of cyclotomic fields positive? [closed]

For any nonzero element in a cyclotomic field $\mathbb{Q}(\zeta_{n})$ with $n \geq 3$, is it true that its norm is positive? If so, how could I possibly show this? This comes from Ireland and Rosen's ...
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Prove that $\mathbb Q(\cos\tfrac\pi7)\neq\mathbb Q(\cos\tfrac\pi9)$

Let $\tau=2\pi$ be the full angle. (tau) For any integer $k$ and any angle $\theta$, $\cos(k\theta)$ is a polynomial in $\cos\theta$. In particular, $\cos(2\theta)=2\cos^2\theta-1$, which shows that $\...
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Intermediate Extensions of $\Bbb Q[\zeta_{63}]/\Bbb Q$.

Let $\zeta$ be the $63^{th}$ root of unity. The reason why I choose the number $63$ is because $$\operatorname{Gal}(\Bbb Q[\zeta]/\Bbb Q)\simeq (\Bbb Z/63\Bbb Z)^\times\simeq(\Bbb Z/2\Bbb Z)\oplus(\...
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Find Intermediate Fields for $\Bbb Q[\zeta_{31}]/\Bbb Q$.

I will very briefly explain the computation. Note that $\phi(31) = 2\cdot 3\cdot 5$ and in $(\Bbb Z/31\Bbb Z)^\times$, one could compute the order of subgroups $\left|\left<5\right>\right|=3$ ad ...
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Compute Intermediate Extensions of $\Bbb Q[\zeta]/\Bbb Q$.

Let $\zeta$ be a primitive $n^{\operatorname{th}}$ root of unity over $\Bbb Q$ then there is a natural isomorphism $$\operatorname{Gal}\left(\Bbb Q\left[\zeta\right]/\Bbb Q\right)\simeq \left(\Bbb Z/...
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Writing the group over cyclotomic fields in GAP [duplicate]

I am able to define the group and do something with this group in GAP. For example, I am interested in the 2I group, and I can write this group in GAP simply as follows: ...
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Criterion for subfield of $ \mathbb{C} $ to be dense

Question: Is it true that a subfield $ K $ of $ \mathbb{C} $ is dense if and only if the roots of unity in $ K $ are dense in the unit circle? Context: I was thinking about the infinite degree ...
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A prime $\mathcal{P}$ dividing $\mathbb{Q}(\zeta)$ divides $\Phi_{m}(\alpha)$

I came across the proof of the following fact in the Chapter $2$, Lemma $2.9$ of Lawrence CWashington's book, Introduction to Cyclotomic Fields. Let $p$ be a rational prime, $n\in\mathbb{N}$ and $a\in\...
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Doubt in Cyclotomic Fields and polynomials

I am reading A Classical introduction to Number Theory by Ireland Rosen. There, they have the following statement and proof: Let $K/\mathbb{Q}$ be an algebraic number field and let $\sigma_1, \...
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Question about automorphisms of the cyclotomic field and Sagemath

Let $K = \mathbb{Q}(\zeta_5)$ be the fifth cyclotomic field. I write the following code in sage ...
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Galois group of an extension generated by character of a group

Let $G$ be a finite group with cardinality $m$ and let $IrrG=\{\chi_1,...,\chi_r\}$ be the set of all irreducible characters of $G$ in $\mathbb{C}$. Since $G$ is finite, then $\chi_i(g)=\sum \zeta_j$ ...
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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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Subfields generated by the rationals and a primitive root of unity.

Let $\zeta_n$ be the primitive $n$th root of unity and $p,q$ be two distinct primes. Then $\zeta_{12} \notin \mathbb{Q}(\zeta_{31})$. If $p$ divides $q-1$ then $\zeta_p\in \mathbb{Q}(\zeta_q)$. If ...
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Any unit in $\Bbb Z[\zeta_p]$ can be decomposed into a power of $\zeta_p$ and a real unit in $\Bbb Z[\zeta_p]$

I am trying to prove the following result, which states that any unit in $\Bbb Z[\zeta_p]$ can be (multiplicatively) decomposed into a power of $\zeta_p$ and a real unit in $\Bbb Z[\zeta_p]$. Let $K =...
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What is known about the minimal absolute value in $\mathbb{Z}[\zeta_n]\setminus\{0\}$?

Here $\mathbb{Z}[\zeta_n]$ is the ring of integers of the cyclotomic field $\mathbb{Q}(\zeta_n)$, where $\zeta_n$ is the $n$th root of unity. In my mind $\mathbb{Z}[\zeta_n]$ looks like a grid in the ...
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Implementation of the cyclotomic numbers

In the Haskell library Cyclotomic, a cyclotomic number is "encoded" as an integer called the order and a map from the integers to the rationals called the coefficients. Would you have an ...
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Sign of image of square root under automorphism of cyclotomic extension

Suppose $m \ne 0, 1$ is a square free integer, $K = \mathbb{Q}(\sqrt m)$ and the ”discriminant” $∆=m$ if $d≡1(mod4)$, and $∆=4m$ otherwise. I am looking for a reference of the fact that the Kronecker ...
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Generator of ideal in $Q(\zeta_5)$

I am thinking about the splitting of 11 in $Q(\zeta_5)$. This totally splits, and I was told that since the minimal polynomial of $\zeta$ completely factorizes mod 11 as $$ x^4 + x^3 + x^2 + x + 1 = (...
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$\left(i\sqrt {\frac{21 \ + \ 5\sqrt{5}}{2}}\right)^n \notin \Bbb{Q}(ζ_{11})$ for all positive integer $n$

I want to prove $\left(i\sqrt {\frac{21 \ + \ 5\sqrt{5}}{2}}\right)^n$ does not lie in $\Bbb{Q}(ζ_{11})$ for all positive integer $n$. This problem arises from arithmetic geometry, but this problem ...
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Question about section 9.1 of Washington's "cyclotomic fields"

I am trying to understand the Basic Argument from Chapter 9.1 of Washington's "cyclotomic fields", and I can understand all but one part. Under Assumption 1: $p \nmid h^{+}(\mathbb{Q}(\...
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Any normal third degree extension of $\mathbb Q$ contained in $\mathbb C$ must be contained in $\mathbb R$

We let $K\subset \mathbb C$ be a finite normal field extension of $\mathbb Q$ such that $[K:\mathbb Q]=3$. Show that $K\subset \mathbb R$. I know that no cyclotomic extension of $\mathbb Q$ will do ...
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On cyclotomics modulo prime powers

Regarding the ring $\mathbb{Z}[X]/\Phi_n(X)$ for the $n$'th cyclotomic polynomial $\Phi_n$, I find much literature on the nature of this ring taken modulo a prime $q$, that is literature on the ...
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The splitting field of $x^{20}-1$ over $\mathbb Q$

I am trying to find the splitting field of $x^{20}-1$ over $\mathbb Q$. I know that it has degree $8$ over $\mathbb Q$, and that $i$ and $\sqrt5$ are in the splitting field. I am also suspecting that $...
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Does $\mathbb{Z}[\zeta_n]$ contain $|x|=1$ with $x$ not a power of $\zeta_n$?

Let $\zeta_n=e^\frac{{2i\pi}}{n}$. Suppose $n$ is even. Then $\mathbb{Z}[\zeta_n]$ contains no roots of unity that are not powers of $\zeta_n$ because this is true of $\mathbb{Q}[\zeta_n]$ which is a ...
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$\mathbb{Q}(\sqrt{-23}) \subseteq \mathbb{Q}(\zeta_{23})$

I want to prove $\mathbb{Q}(\sqrt{-23}) \subseteq \mathbb{Q}(\zeta_{23})$ using the hint from "Introduction to Cyclotomic Fields" Exercise 2.1. $\zeta_{23}$ is the primitive 23-th root of ...
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Prove $1-\xi_p$ is irreducible in $\mathbb{Z}[\xi_p]$

I am trying to prove that $1-\xi_p$ is irreducible in $\mathbb{Z}[\xi_p]$, where $\xi_p$ is a primitive $p$-root of unity. I am not sure how to do this. I have calculated the norm of $1-\xi_p$ and I ...
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$\sigma(\overline{z})=\overline{\sigma(z)}$ with $\sigma$ $\mathbb{Q}$-automorphism

I am trying to prove the following: For fields $\mathbb{Q}\subset F \subset \mathbb{Q}(\xi_n)$, with $\xi_n$ a primitive $n$-th root of unity, prove that $\sigma(\overline{z})=\overline{\sigma(z)}$ ...
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How to prove $x^p+y^p=\displaystyle{\prod_{k=0}^{p-1} (x+\xi_p^ky)}$

I am reading a "proof" of Fermat's last theorem which uses the assumption that $\mathbb{Z}[\xi_p]$ is a UFD (which is actually not true in general). This proof uses the following ...
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Can every algebraic number be written in terms of roots of unity?

Let $ \alpha $ be the root of some polynomial with integer coefficients. Can $ \alpha $ always be written as an algebraic expression using only rational number and roots of unity? This is equivalent ...
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$\xi_p^i-\xi_p^j$ divides $p$ in the ring $\mathbb{Z}[\xi_p]$

I am trying to prove the following: Let $p \geq 5$ be a prime and let $\xi_p\in \mathbb{C}$ be a primitive $p$-th root of unity (that is, $\xi_p\neq 1$ and $\xi_p^p=1$). I want to show that $\xi_p^i-\...
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