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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On some congruence in cyclotomic field

Let $k = \mathbb{Q}(\zeta_5)$ the $5^{th}$ cyclotomic field and $p$ a prime number verify $p\equiv 1 \pmod 5$. Let $\lambda = 1-\zeta_5$ the unique prime of $k$ above $5$, precisely $5 = (1-\zeta_5)^...
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Factoring cyclotomic polynomials in tree form

$\newcommand\ord{\text{ord}_n(q)}$ I've got the following formula for the $n$'th cyclotomic polynomial over the field $\mathbb{F}_q$ ($q=p^r$ and $(n,p)=1$) assuming $q\equiv1(\bmod m)$ as well as ...
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3 votes
2 answers
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Show that $1+\zeta+\dots+\zeta^{k-1}$ is a unit [duplicate]

Let $\zeta=e^{\frac{2\pi i}{p}}$ where $p\geq3$ is a prime. Consider the algebraic number field $K=\mathbb{Q}(\zeta)$. Let $k$ be a positive integer such that $(k,p)=1$ i.e., $k$ is co-prime with $p$. ...
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Identities in maximal real subfields of cyclotomic fields

For the following, let $m=2^{r_{0}}p_{1}^{r_{1}}\cdots p_{n}^{r_{n}}\geq 10$ be an even integer (so that $r_{0}\geq 1$), and let $\omega=e^{2\pi i/m}$ be a primitive $m$-th root of unity. Furthermore, ...
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Quintic residue symbole in the $5^{th}$ cyclotomic field

Let $p_1$ and $p_2$ be prime numbers such that $p_1 \equiv p_2 \equiv \pm2\bmod 5$. We have that $p_1$ and $p_2$ remain inert in the cyclotomic field $F = \mathbb{Q}(\zeta_5)$, where $\zeta_5$ is a ...
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Is it true that $\mathbb{Z}[\zeta_8]/\langle 1+3\sqrt{i}\rangle\cong\mathbb{Z}_{82}$?

I've been thinking a lot about ideals and factor rings, and I came upon the following. First, consider $\mathbb{Z}[\zeta_8]$, which is the cyclotomic ring of integers such that all $z\in\mathbb{Z}[\...
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Algebraic Integers in Cyclotomic fields

Let $\zeta$ be a primitive 8th root of unity in $\mathbb{C}$ and let $\alpha = \frac{1-\zeta}{2}$ (i) Determine the minimal polynomial of $\alpha$ over $\mathbb{Q}$ (ii) Is $\alpha^{-1}$ an algebraic ...
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Material for a first course on cyclotomic fields

I am aware that the two excellent graduate level books on theory of cyclotomic fields are: 1.Introduction to Cyclotomic Fields by Lawrence C. Washington 2.Cyclotomic Fields I and II by Serge Lang Is ...
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Prove $1-ω$ is prime in $\mathbb{Z}[ω]$

Let $p$ be an odd prime and $ω=e^{2\pi i /p}$. Determine if $1-ω$ is prime in $\mathbb{Z}[ω]=\mathbb{Z}+\mathbb{Z}ω+\mathbb{Z}ω^2+...+\mathbb{Z}ω^{p-2}$ My attempt I have tried using the definition of ...
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Existence of a cubic root in a cyclotomic field [closed]

Let $L=\mathbb{Q}(e^{\frac{2i\pi}{9}})$ the $3^2$-th cyclotomic number field and $K=\mathbb{Q}(\sqrt{-3})$ its unique quadratic subfield. Thus, we can consider the element $\alpha = \frac{11+5\sqrt{-3}...
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Sub-Fields of $Q(\zeta_{13})$ with a single generator.

Let $K=Q(\zeta_{13})$ a Number field of degree $12$ over $Q$ which is cyclic and has unique sub-fields $K_2,K_3,K_4$ and $K_6$ of degree $2,3,4$ and $6$ respectively. I need a single generator $\beta\...
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What are the integers that lack unique factorization in $\mathbb Z[\zeta_n],$ $n= 23$? [closed]

In the cyclotomic integers $\mathbb{Z}[\zeta_n]$, $n=23$ has class number $3$ and so unique factorization fails (https://en.wikipedia.org/wiki/Cyclotomic_field ). Can anyone give me examples of such ...
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Finding sine in the real subfield of a cyclotomic field

Consider the cyclotomic field $K=\mathbb{Q}(\zeta)$ where $\zeta = \exp(2\pi i/n)$ is a primitive $n$-th root of unity. I think it's pretty well known that the maximal totally real subfield of $K$ is $...
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How to write an arbitrary element of cyclotomic extension of 6th root of unity?

Let's say $\omega$ is a primitive $6^{\rm th}$ root of unity. Then we know $[\mathbb{Q}(\omega):\mathbb{Q}]=\varphi(6)=2$. If I count the number of linearly independent basis elements for $\mathbb{Q}(\...
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When can a representation of a finite group be defined over certain cyclotomic fields?

Let $G$ be a finite group and $\rho \colon G \to \mathrm{GL}(n, \overline{\mathbb Q})$ be a representation. A theorem of Frobenius says that $\rho(G)$ is conjugate (in $\mathrm{GL}(n, \overline{\...
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How to see that the cyclotomic extension is a separable extension?

$\mathbb{F} \subset \mathbb{E}$ is a cyclotomic extension, if $\mathbb{E}$ is the splitting field of $x^n-1$ over $\mathbb{F}$. We were directly told that $\mathbb{E} \mid \mathbb{F}$ is a Galois ...
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Prove that $N(a+b\xi)=\frac{a^5+b^5}{a+b}$

Let $\xi=e^{\frac{2\pi i}{5}}$, and let $K=\mathbb{Q}(\alpha)$ be a number field. I want to prove that $N(a+b\xi)=\frac{a^5+b^5}{a+b}$, where $a+b\neq0$. If I define the $4$ inmersion like this: $\...
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Maximal abelian $p$-extension unramified away from $p$

I am having trouble seeing how class field theory implies the following isomorphism. Let $F_n=\mathbb{Q}(\mu_{p^{n+1}})$, let $L_p/F_n$ be the maximal unramified abelian $p$-extension (i.e. the $p$-...
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3 votes
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Dedekind zeta function for a subfield of cyclotomic field

Can someone give me a proof or a reference of a proof of Proposition 23 here. I cite it below. $\mu_n$ is an $n$th root of unity. Proposition 23. Let $F$ be a number field contained in $\mathbb{Q}\...
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Upper bound for the cyclotomic number (0,0) of general order over characteristic-2 finite field.

$\mathbb{F}_q$ is a finite field with characteristic 2, namely, $q=2^n$ is a power of 2. $g$ is a generator of the multiplicative group $\mathbb{F}_q^\times=\mathbb{F}_q\backslash\{0\}$. Cyclotomic ...
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The number of solutions to $a+b=1$ in a multiplicative subgroup of a finite field $GF(q)$

I have been wondering what is the number of solutions to $a+b=1$ in an arbitrary multiplicative subgroup $H$ (order $r$) of some finite field $GF(q)$, where $q=p^n$ is a prime power and $a, b \in H$. ...
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Decomposition of unramified primes in normal closure of pure quintic field

Let $\Gamma$ = $\mathbb{Q}(\sqrt[5]{n})$ a pure quintic field, $k$ = $\mathbb{Q}(\zeta_5)$ the $5^{th}$ cyclotomic field, then $N$ = $\mathbb{Q}(\sqrt[5]{n}, \zeta_5)$ is the normal closure of $\Gamma$...
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What effect does the cyclotomic field automorphism $\sigma_a$ have on square roots?

I'm trying to find the effect of $\sigma_a$ on square roots where $\sigma_a$ is defined as an automorphism on the Galois group of a cyclotomic field such that. $$\sigma_a(\zeta_n)=\zeta_n^a$$ Where $\...
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2 votes
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If a number defined by radicals is a root of a polynomial with integer coefficients, then so is the same number taking another branch of a root

Suppose x is a number written with an expression composed of rational constants, sums, subtractions, multiplications, division and extraction of n-th roots. That is, $$x=\alpha^{1/n}\beta+\gamma$$ ...
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If $n$ is odd then $\prod_{j=1}^{\frac{n-1}{2}} (\zeta_n^{t j}+\zeta_n^{-t j}+2)=2^{(t,n)-1}$

Is this proof correct? (Notation: $\zeta_n=\text{exp} (2\pi i/n)$) If $n$ is odd then $\gcd (2,n)=1$. Hence, \begin{align*} \prod_{j=1}^{\frac{n-1}{2}} (\zeta_n^{t j}+\zeta_n^{-t j}+2)&=\prod_{j=1}...
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Finding a square root in a cyclotomic field

Let $x\in\mathbb{Q}[\zeta_{p}]$. Suppose that $$x=a_0+a_1\zeta_p+...+a_{p-2}\zeta_p^{p-2},\quad a_i\in\mathbb{Q}$$ and that $x$ has a square root in $\mathbb{Q}[\zeta_{p}]$, i.e. there exists an ...
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On the representation of $\sqrt{\pm p}$ in the integral basis of $\mathbb Q(\zeta_p)$

I took another look at my previous question on proving a certain trigonometric identity related to the braced heptagon: $$\sin\frac\pi7-\sin\frac{2\pi}7-\sin\frac{4\pi}7=-\frac{\sqrt7}2$$ Around the ...
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Why is $\mathbb{Q}[\zeta_{p}] = \{a_{0}, a_{1}\zeta, \dots, a_{p-2}\zeta^{p-2}, a_{i} \in \mathbb{Q}\}$, where $p \geq 2$ is a prime?

Let $p$ be a prime number $\geq 2$ and $\zeta := \zeta_{p}$ a complex number where $\zeta \neq 1$ and $\zeta^p=1$, i.e., is a $p$th root of unity. This is a question from a brazillian book, Introdução ...
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Understanding Neukirch´s proof

I´m studying algebraic number theory from Neukirch´s book. I´m reading the Proposition 10.2 which says: A $\mathbb{Z}$-basis of the ring $O$ of integers of $\mathbb{Q}(\zeta)$ is given by $1, \zeta, \...
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Norm of generator of prime ideal

Let $K=\mathbb{Q}(\zeta_n)$ be a cyclotomic field. For a prime ideal $\mathfrak{p}$ in the ring of integers $\mathcal{O}_K$, we can write $\mathfrak{p} = (p,f_i(\zeta_n))$, where $\mathfrak{p}\cap\...
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Explicit demonstration that $\mathbb{Z}[\zeta_5]$ is euclidean

I'm reading an introductory book on algebraic numbers. The author has defined the notions of a norm on $\mathbb{Z}[\zeta_n]$ by the usual formula (I give it for $n=5$): $\alpha=f(\zeta_5)=a+b\zeta_5+c\...
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A different approach to $\operatorname{Gal}(\mathbb Q(\zeta_n)/\mathbb Q)\cong(\mathbb Z/n\mathbb Z)^\ast$

As it is well-known we have $\operatorname{Gal}(\mathbb Q(\zeta_n)/\mathbb Q)\cong(\mathbb Z/n\mathbb Z)^\ast$. An approach often taken by textbooks is to first establish the irreducibility and degree ...
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4 votes
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Units in the cyclotomic ring of degree $7$

Let $\zeta=e^{2\pi i \over 7}$. Given $u,v\in\mathbb{Z}[\zeta]$, show that: $$ u\bar{u}+v\bar{v}\,\,\text{ is a unit }\implies (1-\zeta)\mid uv. $$ Motivation An equivalent form of this result is ...
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What is the discriminant of $\Phi_{2}(X)$ over $\Bbb Q$?

What is the discriminant of $\Phi_{2^n}(X)$ over $\Bbb Q$ for $n=1$ since $\Phi_2(X)=X+1$ has only one root $-1$? I have calculated $disc(\Phi_{2^n}(X))$ for $n\geq 2$ which matches exactly with what ...
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How can you specifically extend an automorphism from a quadratic field to one of a cyclotomic field?

I am trying to see how can I extend the automorphisms of the Galois extension $\mathbb{Q}(\sqrt{d})/\mathbb{Q}$, for $d$ square-free, to automorphisms of $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ that fix $\...
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Find all values of the positive integer $n$ such that $Q(e^{\frac {2πi}{n}})$ contains $i.$ [duplicate]

$\mathbf {The \ Problem \ is}:$ Find all values of the positive integer $n$ such that $Q(e^{\frac {2πi}{n}})$ contains $i.$ $\mathbf {My \ approach}:$ I could only think that $4$ divides $n$ as $Q(e^{\...
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Which cyclotomic integers have integer squared modulus?

This feels so elementary, but I have been trying to find examples and have been playing with Wolfram only (where I found some cyclotomic integers with non-integer squared moduli). I am a newbie and ...
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3 votes
2 answers
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For $\omega=e^{2\pi i/3}$ and $\epsilon =e^{2\pi i/5}$, determine $|\mathbb Q[\omega +\epsilon]:\mathbb Q|$.

Let $\omega=e^{2\pi i/3}$ and $\epsilon =e^{2\pi i/5}$, then show the following: (a) $\omega \notin \mathbb Q[\epsilon]$, and (b) determine $|\mathbb Q[\omega +\epsilon]:\mathbb Q|$. Clearly $\omega,\...
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2 votes
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If $\alpha \in \mathbb Q[\zeta_n]$ satisfying $\alpha^m=2$ for some positive integer $m$, then $m=1$ or $2$.

Let $\zeta_n:=$ primitive $n$-th root of $1$ over $\mathbb Q$. And let $\alpha \in \mathbb Q[\zeta_n]$ satisfy $\alpha^m=2$ for some positive integer $m$. Then I have to show that $m=1$ or $2$. We ...
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3 votes
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Ramification in cyclotomic extension

I am trying to understand how primes ramify in $\mathbb Q(\zeta _p)/\mathbb Q$ for $p$ prime. From class field theory, how a prime $q$ ramifies depends only on $q \mod p$. I have following particular ...
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2 votes
2 answers
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Minimal polynomial of $\sqrt[10]{5}$ over $\mathbb{Q}(e^{2\pi i/10})$

Determine the minimal polynomial of $\sqrt[10]{5}$ over $\mathbb{Q}(e^{2\pi i/10})$. Progress so far: 1) Its degree must divide $10$ (since the extension $\mathbb{Q}(\sqrt[10]{5},e^{2\pi i/10})/\...
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1 vote
1 answer
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How to use the Bezout identity in the $mn$-th roots of unity?

Consider $n$, $m$ two natural numbers, and $\zeta_{mn}$ the $mn$-th root of unity. How can I use the Bezout identity in order to prove that there exists $k$ an integer such that $\zeta_{mn}^k$ is a ...
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3 votes
1 answer
123 views

Splitting field of $x^4 + 1$ over $Q$

Let $\alpha$ be a root of $x^4+1$.. so we conclude that all roots of $x^4 + 1$ is primitive 8th root of unity. And the generator is $e^\frac{(2πi)}{8}$. Which is equal to $\cos(2\pi/8) + i \sin(2\pi/8)...
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9 votes
1 answer
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Is the absolute value of the sum of six of the 16th roots of unity ever a nonzero integer?

Let $\zeta_1, \ldots \zeta_{16}$ be the $16$th roots of unity. For the proper subset $J \subset \{1,2,\dots,16\}$ and $|J|=6$, can the following sum ever be satisfied for an integer not equal to zero? ...
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How to prove that $\Pi_{1 \leq j \leq p^n, p \nmid j}((\zeta_{p^n}^j)^{p^{n-1}}-1) = p^{p^{n-1}}$?

How do I prove that $\Pi_{1 \leq j \leq p^n, p \nmid j}((\zeta_{p^n}^j)^{p^{n-1}}-1) = p^{p^{n-1}}$? I calculated some examples in Wolfram Alpha where the results were of this specific form, and it ...
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1 vote
0 answers
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On the quintic power residue symbol

Let $k = \mathbb{Q}(\zeta_5)$ the cyclotomic field, where $\zeta_5$ a primitive $5^{th}$ root of unity. let $p$ a prime integer such that $p \equiv 1 [5]$, more exactly $p \equiv 1 [25]$, then $p$ ...
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show that there exists a unique subfield $H$ of $\mathbb{Q}(\xi_9)$ such that $[H:\mathbb{Q}]=2$.

Let $\xi_9 = e^{\frac{2\pi i}{9}}$, I want to show that there exists a unique subfield $H$ of $\mathbb{Q}(\xi_9)$ such that $[H:\mathbb{Q}]=2$. Is this correct? In that case How one can prove this? I ...
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6 votes
2 answers
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Quadratic subfields of the cyclotomic field $\mathbb{Q}(\zeta_{14})$

In a nutshell, my question is: what is degree of the field extension $\mathbb{Q} \, ( \zeta_{14} + \zeta_{14}^9 + \zeta_{14}^{11}) $ over $\mathbb{Q}$? As to why I'm asking this, I was trying to ...
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3 votes
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Ramification question in compositum of cyclotomic and degree 5 extension.

I am reviewing some past exam questions and I have a problem solving the following example: Let $F = \mathbb{Q}(\zeta_5, \sqrt[5]{75})$, a field of degree 20 over $\mathbb{Q}$. Determine the ...
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How do you find a basis for $\mathbb{C}$ as a vector space over $\mathbb{Q}(\zeta_n)$?

In the comments on a question I posted earlier it was recommended that I find a basis for $\mathbb{C}$ as a vector space over $\mathbb{Q}(\zeta_n)$. How should I go about this? What would be a good ...
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