Mathematics

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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Is $\sqrt 7$ a sum of roots of unity?

Let $a_1,\dots,a_n$ and $b_1,\dots,b_n$ be two sequences of rational numbers. Is it possible that $\sqrt 7 = \sum_{m=1}^{n} a_m (-1)^{b_m}$ ? Is it possible that $\sqrt{17}$ = $\sum_{m=1}^{n} a_m (-1)^{b_m}$ ? How to prove or disprove these ?
mick
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Elementary proof of $\mathbb{Q}(\zeta_n)\cap \mathbb{Q}(\zeta_m)=\mathbb{Q}$ when $\gcd(n,m)=1$.

In an answer to another question I used the fact that $\mathbb{Q}(\zeta_m)\subseteq \mathbb{Q}(\zeta_n)$ if and only if $m$ divides $n$ (here $\zeta_n$ stands for a primitive $n$th root of unity, Edit: and neither $m$ nor $n$ is twice an odd number;…
Álvaro Lozano-Robledo
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Closest cyclotomic integer to a cyclotomic number?

Let's take a cyclotomic field of the form $K=\mathbb{Q}(\zeta_n)$ where $\zeta_p$ is the $n$th root of unity. Then the ring of integers of $K$ is $\mathcal{O}_K= \mathbb{Z}(\zeta_n)$. Is there a generalisation of the rounding function $\left \lfloor…
Chris
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Ramified prime in cyclotomic extension of a number field

Let $K$ be a number field, $n$ be a positive integer and $\zeta_n$ a primitve $n^{th}$ root of unity. How does one show that if a prime ideal $\mathfrak{p}$ of $K$ is ramified in $K(\zeta_n)$ then $n \in \mathfrak{p}$ ? (This is not homework). In…
Zorba le Grec
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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? $$\left\lvert\sum_{j\in J}…
User71942
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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 it is elementary. When $n$ is a multiple of $8$ ,…
Philip Boyle Smith
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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 to asking if $$…
Ian Gershon Teixeira
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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 find the subfields of $\mathbb{Q}\, (\zeta_{14})$ ,…
Adithya Chakravarthy
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Are there any nontrivial unramified extensions between two cyclotomic fields?

Fix $m$ and let $H$ be the Hilbert class field of $\mathbb{Q}(\zeta_m)$ . I'm trying to show that $H\cap \mathbb{Q}(\zeta_n)=\mathbb{Q}(\zeta_m)$ for any $n$ such that $m\mid n$ . To do this, I think that it suffices to show that there are no…
Arbutus
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Minimal polynomial of root of unity over quadratic field

Let $p$ be an odd prime and consider the $p$-th cyclotomic field $\mathbb{Q}(\zeta_p)$ and its quadratic subfield $\mathbb{Q}(\sqrt{\pm p})=:K$. I am interested in the minimal polynomial of a root of unity $\zeta_p$ over $K$ - I understand this may…
Oliver Braun
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Edwards' proof of Fermat's theorem for regular primes

I have been studying the proof of Fermat’s last theorem for regular primes featured in Fermat's Last Theorem : A Genetic Introduction to Algebraic Number Theory (Edwards). This proof of Kummer’s theorem was appealing to me because it doesn’t need…
Tutut
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What's the structure of the sequence of fields $\mathbb Q(\alpha^n)$?

Given an algebraic number $x$ and a natural number $n>0$ , we define the $n$ th powerfield of $x$ as $\mathbb Q(x^n)$ . $$\mathbb Q(x),\; \mathbb Q(x^2),\; \mathbb Q(x^3),\; \mathbb Q(x^4),\; \mathbb Q(x^5),\; \cdots$$ Obviously these are subfields…
mr_e_man
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Parity of the class number of cyclotomic fields

I am interested if there are any heuristics on the parity of the class number of $\mathbb{Q}(\zeta_p)$ where $\zeta_p$ is a primitive root of unity. Is it true that it is odd infinitely many often? Is the density of primes for which it is odd known…
did
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Decomposition and inertia fields in the factorization of $3$ in $\mathbb{Q}(\zeta_{24})$

I've seen the following exercise from an old problem sheet: For $\zeta:=\zeta_{24}$ a primitive $24$ -th root of unity and $\mathcal{O}:=\mathbb{Z}[\zeta]$ , determine the prime decomposition of $3$ . Determine the decomposition and inertia fields…
rmdmc89
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Galois group of $x^n+1$ over $\Bbb Q$

Let $n\in\Bbb Z_{>0}$. Determine the Galois group of $f(x)=x^n+1$ over $\Bbb Q$. I am having some trouble with this. I started by assuming $n$ is odd, then $f(-x)=(-x)^n+1=-(x^n-1)$, then the Galois group of $f(x)$ is the same as $x^n-1$. We know…
Dave
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