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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find algebraic integers in $\mathbb{Q}[x]$ where x is a root of unity

prove that the set of algebraic integers in$\mathbb{Q}[\zeta_p]$ are exactly $\mathbb{Z}[\zeta_p]$where $\zeta_p$ is a primitive p-th root of unity(may assume p is prime) I'm sure that the problem is ...
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Properties of a certain divisor of the $m^{\rm th}$ root of unity in the $m^{\rm th}$ cyclotomic ring.

I am an analyst who is new to algebraic number theory. I am learning on Cyclotomic polynomials and came across this problem, which I am having hard time solving. Let $\xi$ be an $m^{\rm th}$ complex ...
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Generators of a Power of a Prime Ideal

Let $K = \mathbb{Q}(\zeta_n)$, where $\zeta_n$ is an nth root of unity. I know that in $\mathcal{O}_K = \mathbb{Z}[\zeta_n]$, a prime ideal above a given prime $p\in\mathbb{Z}$ has the form $\mathfrak{...
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Equivalence of Cyclotomic units

I am reading Introduction to Cyclotomic Fields by Lawrence C. Washington and I have the doubt regarding the cyclotomic units. Please, see the attached images: In the first image a certain types of ...
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Quadratic number fields that contain primitive root of unity

Find all quadratic fields $\mathbb{Q}[\sqrt{d}]$ that contain some $p$-th primitive root of unity, where $p>2$ is a prime. Now, my reasoning was: if $\mathbb{Q}[\sqrt{d}]$ contains one $p$-th root ...
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Satisfiability of length and phase constraints of three complex numbers

I would like to know if the following question admits a positive or negative answer and if it has been investigated before: Given three distinct powers $\zeta_n^{k_1}, \zeta_n^{k_2}, \zeta_n^{k_3}$, ...
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Beyond angle trisection: Constructing regular polygons by dividing angles into 5, 7, 11, (et cetera) equal parts

I've read a paper by Andrew Gleason where he was able to come up with a way to construct heptagons and tridecagons using angle trisection to supplement the usual compass and straightedge. This post ...
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Fundamental Unit in cylotomic fields

What is the fundamental unit of $Q(\zeta_n)$, where $\zeta_n$ is the primitive $n$-th root of the unity? Is there an explicit algorithm to compute it? I could find https://mathoverflow.net/a/204551/...
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some relation between class number and regulator of cyclotomic field .

Let $K$ be a field totally imaginary number contained in $\mathbb{Q}(\ell^n)$, where $\zeta_{\ell^n} = e^{2i\pi/\ell^n}$ with $n \in \mathbb{N}^*$ and $\ell$ an odd prime number, $F$ the real sub ...
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A question about the main conjecture of Iwasawa theory.

Let $p$ be an odd prime, and $K=\mathbb{Q}(\mu_{p})$. Consider the cyclotomic extension of $K$: $$K\subset K_1\subset K_2\subset\cdots\subset K_\infty ,$$ where $K_n=\mathbb{Q}(\mu_{p^{n+1}})$ and $K_\...
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The classical Gauss sum

This is a problem from dummit and foote 14.7.11 I solved everything except b). I tried hard but couldn't solve it. I'm trying to using sum of pth root of unity is 0. But there are $p$ terms in ...
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$K(\zeta)/k(\zeta)$ is Galois (proof verification)

Let $K/k$ be Galois and $\zeta$ be primitive $n$-th root of unity. I want to prove that $K(\zeta)/k(\zeta)$ is also Galois. Proof: Recall that an extension is Galois iff it is a splitting field of ...
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Exercise 4.8 from Marcus'book “Number fields”

I report here the Exercise with my sketch of a solution. (a) By Exercise 30 Chapter 3, I know that there are infinitely many primes $q \equiv 1 \bmod r$. Moreover, I know that the ring of integers of ...
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Unit roots group is isomorphic to $\Bbb{Q}/\Bbb{Z}\left[\frac{1}{p}\right]$ in a field of characteristic $p\ge0$

Let $K$ be a field so that the group of all unit roots of all orders $\mu_\infty=\bigcup_n {\mu_n}$ (where $\mu_n=\{x\in K\mid x^n=1\}$) splits on $K$. If $K$ is of characteristic $0$, take $p=1$; ...
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Pseudo-isomorphism in Iwasawa Theory

Let $\Lambda$ denote the Iwasawa algebra $\mathbb{Z}_p[[\Gamma]]$, where $\Gamma$ is a group isomorphic to $\mathbb{Z}_p$. We know that $\Lambda$ is homeomorphic to $\mathbb{Z}_p[[T]]$, ring of power ...
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Compute the inertial degrees of two prime ideals e.g the inertial degree of $P/(2)$ where P is prime in $Q[e^{\frac{2\pi i}{23}}]$ lying over (2))

I was reading through Marcus Number Field chapter 3 and I got stuck on exercise 17 Let $K=\mathbb{Q}[\sqrt{-23}]$, $L=\mathbb{Q}[\omega]$ where $\omega=e^{\frac{2\pi i}{23}}$. We know that K $\...
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Why does an element of the Galois group of $K(\mu_n)/K$ maps an $n$-th root of unity to another $n$-th root of unity?

I am working through Keith Conrad's article on Cyclotomic extensions and have a question regarding the proof in Lemma 2.1. Let $K$ be any field and $\mu_n \subseteq K^\times$ be the multiplicative ...
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How to find generators for the subfields of $\mathbb{Q}(\zeta_{12})$

This is somewhat of a follow-up to this question: A complete picture of the lattice of subfields for a cyclotomic extension over $\mathbb{Q}$. After reading this, I am still confused on how to find ...
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Kolyvagin's Euler systems and their properties

I am confused with the different definitions of the Euler system I understand from the book "Cyclotomic Fields and Zeta Values" that an Euler system is a map $φ: W\smallsetminus S \to Q$ (algebraic ...
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Cyclotomic polynomial for any positive integer

i have searched about it but i couldn't get a clear simpler definition of this polynomial, so i need to understand more about it because i am asked how to construct it? And is this polynomial ...
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Difficulty in building a Galois-equivariant map in the cyclotomic case

I am struggling with the proof of the Lemma 2.3 in the appendix of Rubin in the Lang's book "Cyclotomic fields I and II". The setting is the following: $F=\mathbb{Q}(\mu_m)^{+}$, where $m$ is a ...
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Prove that $G(\mathbb{Q}(\zeta)/\mathbb{Q}) $ and $\mathbb{Z}_n^\times$ are Isomorphic

Prove that $G(\mathbb{Q}(\zeta)/\mathbb{Q}) \rightarrow \mathbb{Z}_n^\times$ is an isomorphism. $\mathbb{Q}(\zeta)$ is the cyclotomic extension of $\mathbb{Q}$ ($\zeta$ is a root of $x^n - 1$). $\...
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The constuctability of n-gon(regular n-polygon)

The problem is as follows. Let $p$ is odd prime and suppose that a regular $p$-gon is constructible. Then show that $p$ is a Fermat prime. I solved the $$\Bigg[\Bbb Q\left(\cos\frac{2\pi}{p}\right)...
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How to find the subfields splitting field over $\mathbb Q(i)$ [duplicate]

The question is as follows. Let $L$ is the splitting field of $f(x)=x^{15}-1$ over $\mathbb Q(i)$. Then, find the subfields of $L$ including $K$. How can i find it? I used to find the splitting ...
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Ramification in Maximal Totally Real Subfield

Exercise 12, Chapter 4 of Marcus' Number Fields asks the following: Let $p$ be a prime not dividing $m$. Determine how $p$ splits in $\mathbb{Q}(\zeta_m +\zeta_m^{-1})$. Certainly $p$ is unramified, ...
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Arithmetic in cyclotomic field

I know that the basis of $\mathbb{Z}(\zeta_9)$ is: $\{1,\zeta_9,\zeta_9^2,\zeta_9^3,\zeta_9^4,\zeta_9^5\}$ . If I have for example: $3+104\zeta_9+\zeta_9^7$ . Is it a cyclotomic integer in $\...
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Cyclotomic field of 9th root of unity

What I know: The set $\{1,\zeta,\,...\,,\zeta^{\varphi(n)-1}\}$ is an integral basis for the ring of algebraic integers of $\mathbb{Q}(\zeta_n)$ . Then, if: $\mathbb{Q}(\zeta_9)$ . As $\varphi(...
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The Ideal Class Group of $R = \mathbb{Z}[e^{2i\pi/p}]$

I wish to show that ideal class group of $R$ is finite. To accomplish this I am given several parts to prove. I have proven some of them but not all. These are the things I am stuck on. Suppose A is ...
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Prove that $(1 - \omega)^{\phi(m)} = p$ where $\omega = e^{2\pi i/m}$

I was reading Marcus' Number Fields and came across a line saying "we know $(1 - \omega)^{\phi(m)} = p\mathbb{Z}[\omega]$" where $\omega = e^{2\pi i/m}$ and $m = p^r$. Earlier on in the book it ...
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Evaluate sum roots of unity $\zeta_{17}+\zeta_{17}^{-1}+\zeta_{17}^{2}+\zeta_{17}^{-2}+\zeta_{17}^{4}+\zeta_{17}^{-4}+\zeta_{17}^{8}+\zeta_{17}^{-8}$

For $\zeta_{17}$ a primitive $17^{\text{th}}$ root of unity, I need to reduce $\zeta_{17}+\zeta_{17}^{-1}+\zeta_{17}^{2}+\zeta_{17}^{-2}+\zeta_{17}^{4}+\zeta_{17}^{-4}+\zeta_{17}^{8}+\zeta_{17}^{-8}$ ...
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Minimal polynomial of $\eta_4$ over $\mathbb{Q}(\eta_2)$.

I know that for each divisor $d \mid 17-1=16$ there is a unique subfield $K_d ⊆ \mathbb{Q}(\zeta_{17})$ with $[K_d : \mathbb{Q}] = d$. Moreover, for $H_d ≤ (ℤ/17ℤ)^*$ the subgroup of index $d$, and $\...
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Proving that $\mathbb{Z}[e^{2i\pi/23}]$ is not a PID

Some of the proofs of this claim involves showing that 47 cannot be a norm of any element in $\mathbb{Z}[e^{2i\pi/23}]$. I see that this implies that any ideal of norm 47 cannot be principal, but how ...
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On the irreducibility over cyclotomic field in radical extension.

I'm interested in the splitting field $\mathbb{Q}(\sqrt[20]{2},\zeta_{20})$ of polynomial $f(x):=x^{20}-2\in\mathbb{Q}[x]$ over $\mathbb{Q}$, where $\zeta_{20}=e^{\tfrac{2\pi i}{20}}$. Clearly, $f(x)$ ...
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Algebraic Integers of a Field

Suppose my ring is $\mathbb{Z}[e^{i2\pi/p^2}]$. I wish to prove that the algebraic integers are ${\sum_{n = 0}^{p^2-p-1}a_{n}e^{i2n\pi/p^2}}$ where each $a_{k} \in \mathbb{Z}$. I see why thus finite ...
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Group of Units a Subgroup of some ${S}_n$

Consider the polynomial $f(x) \in \mathbb{F}[x]$ where $\mathbb{F}$ is some field of characteristic 0. Let $\mathbb{E}$ be the splitting field of $f$. We know that if $f$ is separable of degree $n$, ...
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Factoring Primes in cyclotomic fields

I have some questions regarding factoring a prime in a cyclotomic field. Say I want to factor 241 in $\mathbb{Z}[\zeta_3]$, I know that I am supposed to look for roots of $x^2+x+1$ in $\mathbb{Z}_{241}...
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The action of a Galois group on a cyclotomic field.

Let $\zeta_5$ be a primitive $5^{th}$ root of unity, $k=\mathbb{Q}(\sqrt[5]n,\zeta_5)$ the normal closure of $\Gamma = \mathbb{Q}(\sqrt[5]n)$, ${\rm Gal}(k/\Gamma)= <\tau>$ with $\tau : \zeta_5 ...
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Congruence in cyclotomic fields

Let $\Gamma = \mathbb{Q}(\sqrt[5]n)$ be a purely quintic field with $n = 5^ep^{e_1}$ and $p$ a prime number verify $p\equiv 1 [5]$ and $p\not\equiv 1 [25]$. Let $k_0 = \mathbb{Q}(\zeta_5)$ the $5^{...
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Fermat's Little Theorem in cyclotomic fields

Suppose Fermat's Little theorem is generalized to $q$-th cyclotomic polynomials (for $q$ prime) in the following manner: $$(x+a)^n=x^n+a = x^{(n \bmod q)}+a \pmod {\Phi_q(x),n}$$ if $n$ is prime. ...
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subfields of the cyclotomic fields $\mathbb{Q}(\zeta_p)$ with $p \equiv 1 [5]$

Let $p\equiv 1 (mod 5)$ a prime number, let $\mathbb{Q}(\zeta_p)$ the cyclotomic field of degree $p-1$ over $\mathbb{Q}$, its known that $\mathbb{Q}(\zeta_p)$ has unique subfield of degree $5$. My ...
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Proof of a relation of Stickelberger element

I am reading the book Introduction to Cyclotomic Fields by Lawrence C. Washington and I am stuck in proving the relation $\varepsilon_i \theta = \frac{1}{p} \sum_{a=1}^{p-1} a{\omega}^{-1}(a)\...
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Why in cyclotomic extension, Gal($K/F$) may not isomorphic to the whole $Z/nZ$?

In this proof (picture below), the writer defines a map $\theta$, and prove it to be homomorphism and injective. Thus Gal$(F(w)/F)$ is isomorphic to a subgroup of ${Z_n^*}$. Why it cannot be '...
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Proof of a relation of elements in group rings

I am reading the book Introduction to Cyclotomic Fields by Lawrence C. Washington and I need to prove certain identities for the orthogonal idempotents of a group ring (namely a,b,c,d in the attached ...
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Show that $\text{tr}(x(1-\zeta)) \in (1-\zeta)\mathcal{O}_K$

This is a part of proof showing that the ring of integer for $K=\mathbb{Q}(\zeta)$ is $\mathbb{Z}[\zeta]$, where $\zeta$ is a $p$-th root of unity. Show that if $x=a_0+a_1\zeta+\cdots+a_{p-2}\zeta^...
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class group size of cyclotomic field subextension

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{...
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Find all the fixed subfield of $Gal(\mathbb{Q}(\zeta_{40})/\mathbb{Q})$

Given the polynomial $$p(x) = x^{8}+x^{7}+x^{6}+x^{5}+2x^{4}+x^{3}+x^{2}+x+1$$ The task was to find the splitting field,a primitive element of the extension, the Galois group,including counting all ...
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Inertia group of an extension of Local Fields

I am going through a proof in Chapter V of Neukirch's Algebraic Number theory, and would require some clarifications. The theorem we are trying to prove is that the local version of the Kronecker-...
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Roots of unity in $\mathcal{O}_{K}$, where $K=\mathbb{Q}(\sqrt{5},\sqrt{-2})$.

Let $K=\mathbb{Q}(\sqrt{5},\sqrt{-2})$. I already showed that the ring of integers in $K$ is $\mathbb{Z}(\frac{\sqrt{5}+1}{2},\sqrt{-2})$ and the discriminant is $1600$. Now the goal is to show that ...
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Why can I claim that one of these ideals must be 1 in a proof regarding primality of ideals for cyclotomic fields?

In Washington's Introduction to Cyclotomic Fields, we are given the following lemma (1.4): The ideal $(1 - \zeta)$ is a prime ideal of $\mathscr{O}$ and $(1 - \zeta)^{p-1}$= (p). Therefore p is ...
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Some Galois groups of cyclotomic field.

Let $p$ be an odd prime, $\zeta_p=e^{{2\pi i}/p}\in\mathbb{C}$, $F=\mathbb{Q}(\zeta_p)$. Let $E\subseteq\mathbb{C}$ be the splitting field of $X^p-2$ over $\mathbb{Q}$. Let $F_0=F\cap\mathbb{R}$ ...

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