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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31 views

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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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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1answer
38 views

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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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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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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1answer
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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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1answer
62 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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228 views

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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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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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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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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Proof of Iwasawa's theorem

I am reading Iwasawa's theorem from Washington's Cyclotomic Fields. The theorem goes as follows: I do not understand the third line of the proof. Why is $\text{Gal}(L_n/K_n) \simeq A_n$? Any help is ...
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Compute some of powers of root of unity in $\Bbb C$

Let $q=p^r$ be an odd prime power. Let $\epsilon \in \Bbb C$ be a primitive $(q-1)$-th root of unity and let $d\in \Bbb Z_{\ge 1}$. I want to prove that $\sum\limits_{k=1}^{\frac{q-3}{2}} (-1)^k (\...
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$e^{2\pi i/k}\mapsto e^{2\pi i m/k}$ automorphism on the cyclotomic field $\mathbb{Q}(e^{2\pi i/k})$

I am currently reading a proof in which they mention $e^{2\pi i/k}\mapsto e^{2\pi i m/k}$ as an automorphism on the cyclotomic field $\mathbb{Q}(e^{2\pi i/k})$. I know next to nothing about cyclotomic ...
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Algorithm for computing residues of Galois conjugates of an integer in a prime-power cyclotomic field

Let $l$ be an odd prime and let $L=\mathbb{Q}(\zeta_{l^n})$. Suppose $x,y\in \mathbb{Z}[\zeta_{l^n}]$. Let $\mathfrak{p}\mid p$ be a split prime (so in particular $p\equiv 1 \ (\text{mod} \ l^n)$) so ...
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Intersection of cyclotomic fields over an arbitary number field

Let $K$ be an arbitrary number field and $m,n$ are two co-prime integers. Suppose we know that, $[K(\zeta_m):K]=\phi(m), [K(\zeta_n):K]=\phi(n).$ Then is it true that $[K(\zeta_{mn}):K]=\phi(mn)?.$ I ...
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Do there exist elements in cyclotomic fields with certain absolute value?

Consider the $m$-th cyclotomic field $K$ over $\mathbb Q$. For an integer $n\ge2$, we know it is possible that there does not exist integer $\alpha\in\mathcal O_K$ such that $\alpha\bar\alpha=n$. What ...
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Cyclotomic units modulo $p$-th powers in the cyclotomic tower

Let $E$ be the unit group of the cyclotomic field $L_n=\mathbb{Q}(\zeta_{p^n})$ where $\zeta_{p^n}$ is a primitive $p^n$-th root of unity. Let $G=Gal(L_n/\mathbb{Q})$ and let $\Delta=Gal(L_1/\mathbb{Q}...
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69 views

Proof of a theorem in Washington's book on Cyclotomic fields.

I am reading chapter 13 - 'Iwasawa theory of $\mathbb Z _p$-extensions' in Washington's book 'Cyclotomic fields'. On page 267 it is claimed that for extension $F$ (which is infinite as it is maximal ...
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Section : Cyclotomic extension Hungerford Algebra

This question is from Algebra by Thomas Hungerford page 301 (question 5) and I was unable to solve it. Let $g_n (x) $ be cyclotomic polynomial over $\mathbb{Q}$. If p is a prime and $k \geq 1$ then ...
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49 views

How to show $\zeta_p\in K$

Let $p$ be a prime and let $K$ be a field of characteristic $d$ co-prime to $p$. Let $a\in K$. I want to show, $a\in (K^*)^p\iff a\in (K(\zeta_p)^*)^p$, and this would eventually imply $\zeta_p\in K$. ...
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Dot Product of Vectors of Roots of Unity

Let $n \in \mathbb{N}$ and $i$ be the usual complex number such that $i^2=-1$. Let $\zeta = \exp(\frac{\pi}{n} i)$, $v = [1,\zeta,\zeta^2,\ldots,\zeta^{n-1}]$. Given $J \subseteq \{1,\ldots,n\}$, let $...
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Trace map from $\Bbb Q\Bbb(\zeta_r)$ to $\Bbb Q\Bbb.$

I am interested in knowing how to get values of $Tr_r(a)$. Where $Tr_r$ is trace map from $\Bbb Q\Bbb(\zeta_r)$ to $\mathbb Q\mathbb,$ and $\zeta_r$ are some specific complex roots of unit. Not really ...
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Ramification in subfield lattice of a cyclotomic field

Fix a positive integer $n=p_1^{a_1}\cdots p_k^{a_k}$. Consider the cyclotomic field $K=\Bbb Q(\zeta_n)$. I know that only $p_1,\ldots,p_k$ are ramified in $K/\Bbb Q$, and that the ramification index ...
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When is $\sqrt{2}$ in $\mathbb{Q}(\zeta_n)$?

The following question cropped up in a physics context when studying some properties of conformal field theory. I know a little Galois Theory, but not enough to be able to answer it, though I suspect ...
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48 views

Quotient ring Isomorphism in dedekind domain

I was reading Jurgen Neukirch and I came across this in lemma $10.1$ and $10.2$ Let $\zeta_n$ be the $n$th root of unity and $n=l^k$, $l$ being a prime, then $(\lambda)=(1-\zeta_n)$ is a principle ...
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Artin symbol in cyclotomic fields

I'm trying to convince myself that when $p\nmid m$, $p$ prime, then $\left(\frac{\mathbb{Q}(\zeta_m)/\mathbb{Q}}{p}\right)$ is the map $\zeta_m\mapsto \zeta_m^p$. I'm sure it's both true and obvious, ...
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1answer
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How do I represent the elements of $\mathbb{F}_9$ over $\mathbb{F}_3$

How do I represent the elements of $\mathbb{F}_9$ over $\mathbb{F}_3$? The question hints to use the fact that $\mathbb{F}_9$ is the 8th cyclotomic field over $\mathbb{F}_3$, but I dont see how that ...
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A question regarding reason behind assumptions in 2 proofs of Cyclotomic extensions

I have self studied section Cyclotomic extensions( page 298 ) from Algebra by Thomas Hungerford and I have 1 question Consider these 2 theorems : Question: why does in both the theorems there is an ...
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2answers
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What is meant by dimension of extension in Field Theory

I am self studying Field Theory from Thomas Hungerford and I am having confusion in understanding meaning of these 2 terms in section " Cyclotomic Fields" ( page 297). What is meant by : In ...
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Lower for the inertial degree $f$ in cyclotomic field

Let $\mathbb{Q(\zeta)}/\mathbb{Q}$ be a galois extension of degree $p-1$ where $\zeta=e^{\frac{2 \pi i}{p}} $ Let $\mathfrak{b}$ be an ideal above $q$ Where q is a prime in $\mathbb{Z}$ different from ...
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1answer
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On the factorization of prime ideals of $\mathbb{Z}$ in $\mathbb{Z}[\zeta]$

Let $\mathbb{Q(\zeta)}/\mathbb{Q}$ be a galois extension of degree $p-1$ where $\zeta=e^{\frac{2 \pi i}{p}} $ and let $G=\operatorname{Gal}(\mathbb{Q(\zeta)}/\mathbb{Q})$ be its Galois group. Suppose ...
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1answer
40 views

Degree of the $n$-th cyclotomic extension

I have this proposition in the book I'm studying: Consider $n>2$ on $\mathbb{N}$ and $p$ a prime number such that $p\nmid n$ and $p$ is not equivalent to $1$ modulo $n$. If the factorization of $n$...
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On the factorization of ideals in a subfield of cyclotomic extension

Let $m,n$ be integers s.t $m$ divides $n $ The Kronecker-weber theorem tells us that for any ablien group G of order n there is Galois extension L over $\mathbb{Q}$ s.t $ L \subset \mathbb{Q(\zeta_{l}...
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1answer
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Need help in verifying a map to be monomorphism

While studying algebra from Thomas Hungerford I have a question in proof of Theorem 7.11 on page 298. It's image: In last fifth line of proof I am not able to verify how the map $\sigma -> \bar {i}...
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1answer
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Galois group of $\mathbb{Q}(\zeta_p)/\mathbb{Q}(\sqrt{p})$

I just started to study Galois theory and so I'm not too good with calculating Galois groups, I know that $\operatorname{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q})=(\mathbb{Z}/p\mathbb{Z})^\times$ but what ...
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Understanding action of $(\mathbb Z_p/(p^n))^ \times$ in Lubin Tate formal group laws context on $\mathbb{Q}_p[\zeta_{p^r}]$

I want to understand that the isomorphism $$(\mathbb Z_p/(p^n))^\times\ \stackrel{\sim}{\longrightarrow}\ \text{Gal}(\mathbb Q_p[\zeta_{p^r}]/\mathbb Q_p),$$ is the standard one. So how does element $...
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Artin conductor of cyclotomic extensions.

Let ${\Bbb Q}_p(\zeta_{p^n})/{\Bbb Q}_p$ be a cyclotomic abelian extension of degree $p^n - p^{n-1}$ and define $G \colon= {\mathrm{Gal}}({\Bbb Q}_p(\zeta_{p^n})/{\Bbb Q}_p)$, for which we set $G = \...
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Find the minimal polynomial of the 18th root of unity and the Galois group of $\mathbb{Q}(z_{18})$ and its generators [duplicate]

Find the minimal polynomial of the 18th root of unity and the Galois group of $\mathbb{Q}(z_{18})$ So the minimal polynomial is $x^6-x^3+1$ and i checked online and it is correct. Now for the galois ...
2
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63 views

Euler Systems and Coleman’s Conjecture

I’m trying to work on Coleman’s conjecture for abelian extensions of imaginary quadratic fields. I’ve read most papers by Seo regarding circular distributions. However, I’m a still confused about what ...
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52 views

Find the Subfield diagram of splitting field of $x^{33}-1$ over $\mathbb{Q}$

Find the subfield diagram of splitting field of $x^{33}-1$ over $\mathbb{Q}$ This problem seems hard. I cannot find the fields using the automorphisms, because the basis is too large and so it would ...
2
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59 views

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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25 views

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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69 views

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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28 views

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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