# 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.

307 questions
Filter by
Sorted by
Tagged with
31 views

25 views

### 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 ...
84 views

27 views

### 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 ...
62 views

20 views

### $e^{2\pi i/k}\mapsto e^{2\pi i m/k}$ 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 ...
21 views

### 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 ...
82 views

### 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 ...
66 views

### 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 ...
69 views

49 views

### 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 ...
50 views

### 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 ...
89 views

### 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 ...
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 ...
36 views

### 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, ...
38 views

### 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 ...
21 views

### 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 ...
52 views

### 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 ...
55 views

### 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 ...
83 views

### 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 ...
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 $n$...
27 views

76 views

### 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 ...
67 views

26 views

### 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 ...
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 ...
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 ...
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 ...
### 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 ...