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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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Totally $p$-adic elements in cyclotomic extensions of $\mathbb{Q}$

Let $p$ be a prime number and fix $\bar{\mathbb{Q}}$ an algebraic closure of $\mathbb{Q}$. An element $\alpha \in \bar{\mathbb{Q}}$ is called totally $p$-adic if $p$ splits completely in $\mathbb{Q}(\...
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Isomorphism $\mathbb Z[\omega]/(1-\omega)^2\cong (\mathbb Z/(p))[x]/(1-X)^2$, $\omega$ is the $p-$th root of unity.

Im reading the following proof of Fermat's Last Theorem from Keith Conrad https://kconrad.math.uconn.edu/blurbs/gradnumthy/fltreg.pdf On page 5 he mentions that $\mathbb Z[\omega]/(1-\omega)^2\cong (\...
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Galois Group of $\mathbb Q$ Surjects onto Certain Cyclotomic Extension

Let $\ell$ be a prime and let $G_{\mathbb Q, \ell}$ denote the Galois group of $\mathbb Q(\mu_{\ell^\infty})/\mathbb Q$, the extension of $\mathbb Q$ formed by adjoining all primitive $\ell^n$th roots ...
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Linear Dependence of Primitive Roots of Unity

Consider the cyclotomic field $\mathbb{Q}(\zeta_n)$. We know that the set of primitive roots $\Pi_n=\{\zeta_n^m:(m,n)=1\}$ generates $\mathbb{Q}(\zeta_n)$ as a field. However, what happens when we ...
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$\sqrt{p^*}$ contained in $\Bbb Q(\zeta_p)$ [duplicate]

Let $p$ be a prime. I read several times e.g. here that the square root $\sqrt{p^*}$ (where $p^*:=(-1)^{(p-1)/2}p$) is contained in $\Bbb Q(\zeta_p)$. Is there a "standard argument" see it? ...
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Check when a positive square Root $\sqrt{d}$ is contained in Cyclotomic Field

Let $\zeta_n $ be a primitive root of unity generating the cyclotomic field $\Bbb Q(\zeta_n)$. Is/are there quick and/or "standard" techniques" to check if a given real quadratic roots $...
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Mod $\ell$ Cyclotomic Character Evaluated at Frobenius Away From $\ell$

Let $\ell$ be a prime, let $p \neq \ell$ be prime, and let $P$ be a prime lying above $p$ in $\overline{\mathbb Z}$, the ring of integers of $\overline{\mathbb Q}$. If $\chi_\ell$ is the $\ell$-adic ...
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Why should a splitting field of a polynomial over $\mathbb{F}_p[X]$ be a cyclotomic extension?

I have just been doing a Galois theory exercise and one part of the exercise requires me to explain why, if $f \in \mathbb{F}_p[X]$ is a polynomial of degree $n$, its splitting field $L$ is a ...
Featherball's user avatar
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Calculate explicitly $a$ satisfying $(1-\zeta_p)^{p-1}= pa$ inside $\mathbb{Z}[\zeta_p]$

Let $p \neq 2$ be a prime and $\zeta_p$ be a (nontrivial) root of unity. It is well known that $(1-\zeta_p)^{p-1}$ is divisible by $p$ inside $\mathbb{Z}[\zeta_p]$, i.e., there exists $a \in \mathbb{Z}...
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A Formula for the Norm in the Cyclotomic Field of Degree 5

So you all know and love, the Gaussian integers have a rather neat-looking norm: given $a+bi$, then $N(a+bi)=a^2 + b^2$. When it comes to the cyclotomic integers of degree 3, i.e. the domain $\mathbb{...
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The primitive $n^{th}$ roots of unity form basis over $\mathbb{Q}$ for the cyclotomic field of $n^{th}$ roots of unity iff $n$ is square free

Prove that the primitive $n^{th}$ roots of unity form a basis over $\mathbb{Q}$ for the cyclotomic field of $n^{th}$ roots of unity if and only if $n$ is square free I think I have the $(\Rightarrow)$...
Grigor Hakobyan's user avatar
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How to calculate Gal$(F(\mu_{p^\infty})/F(\mu_p))$ for a number field $F$?

Let $F$ be a number field. Recall that we define $$F(\mu_{p^\infty})=\bigcup_{n=1}^{\infty}F(\mu_{p^n}).$$ I want to calculate the group Gal$(F(\mu_{p^\infty})/F(\mu_p))$. I know that this is supposed ...
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When is $\sqrt n$ in $\Bbb Q[\omega_m]$?

Given a positive integer $n$ and a primitive $m^{th}$ root of unity $\omega_m$ over $\Bbb Q$, how could one determine if $\sqrt{n}$ lies in $\Bbb Q[\omega_m]$? In the case of $n=p>0$ being an odd ...
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If $\sqrt{n}$ is contained in $\Bbb Q[\omega_m]$, so does $\sqrt{p}$ for any $p|n$.

I know that every square root $\sqrt{n}$ of an integer is contained in some cyclotomic extension $\Bbb Q[\omega_m]$. If we know $\sqrt{n}\in\Bbb Q[\omega_m]$ and that $n$ is square-free, can we ...
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Why is $K \cap \mathbb{Q}^{\text{cyc}}=\mathbb{Q}$ iff $\chi_K(G_K)=\hat{\mathbb{Z}}^{\times}$?

I am reading David Zywina's "Elliptic curves with maximal Galois action". For a number field $K$, he defines $\mathbb{Q}^{\text{cyc}} \subset \overline{K}$ to be "the" cyclotomic ...
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When is $\sqrt p$ contained in $\Bbb Q[\omega_n]$?

Given any positive prime $p$, there is some cyclotomic extension $\Bbb Q[\omega_n]$ of $\Bbb Q$ containing $\sqrt p$ as a consequence of Kronecker–Weber theorem. But more specifically, given any ...
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Is $\bigg\lvert\frac{1 - \zeta^s }{ 1 - \zeta}\bigg\rvert \in\mathbb{Q}(\zeta)$ for $s = 1, \ldots, p-1$?

Let $p$ be an odd prime and let $\zeta$ be a $p$-th root of unity. Is it true that $$\Bigg\lvert\frac{1 - \zeta^s }{ 1 - \zeta}\Bigg\rvert \in\mathbb{Q}(\zeta)$$ for $s = 1, \ldots, p-1$? Remark: By $\...
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Proving $\mathbb{Q}(\sqrt 2) \subset \mathbb{Q}(\zeta_n) \iff n = 8 \cdot m$ for $ m \in \mathbb{N}$

My ultimate goal is showing that $\mathbb{Q}(\sqrt 2) \subset \mathbb{Q}(\zeta_n) \iff n = 8 \cdot m$ for $ m \in \mathbb{N}$. I have showed that for $n= 8 \cdot m$ the result holds. There is only one ...
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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 ...
Handle135790's user avatar
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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 ...
eagle I 's user avatar
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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 ...
zukoatpeace's user avatar
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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(\...
William Sun's user avatar
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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 ...
William Sun's user avatar
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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/...
William Sun's user avatar
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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\...
HumbleStudent's user avatar
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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 ...
Ninja's user avatar
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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 $\chi_i(g)=\sum \zeta_j$ ...
Mario's user avatar
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2 answers
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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 =...
stoic-santiago's user avatar
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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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