19
votes

Accepted

### Why is the regulator of the degree 11 extension of $\mathbb{Q}$ so large?

A prime $\ell$ splits in $K_p$ for $p$ odd if and only if $$\ell^{p-1} \equiv 1 \mod p^2.$$
We can express $\lim_{s \to 1} (s-1) \zeta_{K_n(s)}$ as the product of $\frac{1}{ 1- |\mathfrak a|^{-1} }$ ...

- 119k

15
votes

Accepted

### Special units in the $11$th cyclotomic field

Yes. Indeed
$$
(1 + \zeta + \zeta^{10}) \, (\zeta + \zeta^4 + \zeta^7 + \zeta^{10}) = 1
$$
with $\sum_i a_i = 3$ and $\sum_i b_i = 4$; now change each $a_i$ to $a_i+3$
and each $b_i$ to $b_i+3$. (...

- 72.6k

13
votes

### Can a sum of roots of unity be an integer?

If $n$ is square-free this cannot happen (even just for the case $k=n$), and if $n$ is not square-free (and in the case $k=n$) one must have the sum being zero if it is an integer (as seen in Peter ...

- 42.6k

12
votes

Accepted

### Points of elliptic curves over cyclotomic extensions

$E (\mathbb Q^{ab})/\operatorname{tors}$ is a sum of countably many copies of $\mathbb Z$.
To prove this, take a countable basis $x_1,x_2,\dots$ of $E (\mathbb Q^{ab})/\operatorname{tors}\otimes \...

- 119k

12
votes

### Can a sum of roots of unity be an integer?

We suppose towards a contradiction that all the sums in the first question are integers. Set $$f(x) = \prod_{h \in H}(x - \zeta_n^h)$$
We claim that all symmetric polynomials in the roots of $f$ are ...

- 11.1k

12
votes

Accepted

### Can a sum of roots of unity be an integer?

Below is the original and accepted proof, but see the by a factor of $100$ simpler later proof in the OP's answer!
The answer is yes for the second question, with $n=8$, since $\zeta_8+\zeta_8^5=0$.
...

- 16.3k

12
votes

### Products of cyclotomic polynomials

Try $x^{30}+x^{20}+x^{15}+x^{10}+1=\Phi_6\Phi_{30}\Phi_{25}$. For more general solutions see the extended comment below.
One way to search for such examples is to solve for the set of polynomials ...

- 15.4k

11
votes

Accepted

### Which criteria for "uniformly splitting" polynomials?

Yes, there are uniformly splitting polynomials of all degrees which are not of the form $\Phi_{n}(x^{k})$. (For example, $f(x) = x^5 + x^4 - 4x^3 - 3x^2 + 3x + 1$.)
The Chebotarev density theorem ...

- 18k

10
votes

### Points of elliptic curves over cyclotomic extensions

Since you ask more generally for results on $E(\mathbb{Q}^{\mathrm{ab}})$, let me expand my comment into a short answer.
Amoroso and Dvornicich discovered (A lower bound on the height in abelian ...

- 13.5k

9
votes

### Regulator of abelian extensions of Q

All of your questions have comprehensive and beautiful answers: providing them is exactly what the subject of Iwasawa theory was developed to do. You should perhaps read either Washington's Cyclotomic ...

- 32.6k

9
votes

Accepted

### A sum involving roots of unity

Here is the proof of Kevin Liu's version
$$
\sum_{k=1}^{2n+1}\left(\frac{(-y)^k}{1-y^{3k}}+ \frac{y^{k}}{1+y^{3k}}\right)=-n-1
$$
(for the primitive root of unity $y$ of degree $6n+4$) of Nemo's ...

- 90.4k

9
votes

Accepted

### Products of Cyclotomic Polynomials with Nonnegative Coefficients

You might do well to try more specific cases. I'll give you a result on the case you end with and a few more showing why I think the general case might be unmanageable.
Claim: $\Phi_2^i\Phi_3^j\Phi_6^...

- 29.6k

9
votes

### Can a sum of roots of unity be an integer?

EDITED Despite the upvotes, the first version of this answer was mostly wrong, and I cannot delete it since it has been accepted. Let me see what can be salvaged.
Fix $\zeta_n$, identify $(\mathbb{Z}...

7
votes

Accepted

### Motivation for cyclotomic units

I'm quite sure cyclotomic units were first introduced by Kummer in his 1847 paper, where he proved his very famous partial solution to Fermat's Last Theorem:
Theorem (Kummer) If $p$ is an odd prime ...

- 17.1k

6
votes

### Special units in the $11$th cyclotomic field

If I did this right there are a total of $1045 = 55 \cdot 19$
solutions, obtained from the following $19$ basic solutions by changing
$a_i,b_i$ to $a_{ri+s\bmod 11}$ and $b_{ri+s \bmod 10}$ for all
$s\...

- 72.6k

6
votes

Accepted

### Is there a pattern for the irreducible factors and degrees of a characteristic polynomial?

I made hand calculations for $n\le4$. It turns out that even with $V_n=1_n$, the factor $U_n$ is not unique. In other words, the factorisation problem is underdetermined. This is due to the fact that ...

- 48k

6
votes

### The average number of solutions to $a+b=c$ in a multiplicative subgroup of $\mathbb{F}_q^\times$ when $c\in\mathbb{F}_q^\times$ is random

The average number of solutions is equal to the sum over all $c \in \mathbb F_q^\times$ of the number of solutions divided by $q-1$. Thus, it is equal to the number of pairs, $a,b \in H$ such that $a+ ...

- 119k

5
votes

### On largest degree of polynomial related to cyclotomic polynomials - I

A trivial answer (to 2) is just that if $f(x),g(x) \in \mathbb{Z}[x]$ have degrees $m,n$ and coefficients in $\{-1,0,1\}$ then $f(x)g(x) = \sum_{k \leq m+n} x^k \sum_{i+j=k;i \leq m;j \leq n} f_i g_j$ ...

- 601

5
votes

Accepted

### Splitting of primes in cyclotomic $\mathbb{Z}_p$-extension

For any $\mathbb{Z}_p$-extension the ramification is concentrated among the primes above $p$. Those that are ramified are totally ramified.
For the cyclotomic $\mathbb{Z}_p$-extension all places ...

- 7,014

5
votes

### Realizability of a real representation using real cyclotomic coefficients

$\def\CC{\mathbb{C}}\def\RR{\mathbb{R}}\def\QQ{\mathbb{Q}}\def\ZZ{\mathbb{Z}}$The main result of this answer will be that, if $V$ is a representation defined over $\RR$ and over $\QQ(\zeta_M)$, then ...

- 141k

5
votes

### Classification of cyclotomic fields with class number 1

The complete list of $n$ for which ${\bf Q}(\zeta_n)$ has unique factorization is
1 through 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 50, 54, 60, 66, 70, 84, 90.
Source; ...

- 51

4
votes

### Algebraic numbers abhorrent to cyclotomic fields

This type of problem arises in Dobrowolski's famous result [1] on Lehmer's conjecture. Here is a result from Dobrowolski's paper:
Lemma 3. Let $\alpha$ be an algebraic number of degree $n$. Then
$$
\...

- 42.7k

4
votes

Accepted

### Something interesting about the quintic $x^5 + x^4 - 4 x^3 - 3 x^2 + 3 x + 1=0$ and its cousins

I think the depressed quintic in the question is what Emma Lehmer called the reduced quintic in her paper, The quintic character of 2 and 3, Duke Math. J. Volume 18, Number 1 (1951), 11-18, MR0040338. ...

- 36.5k

4
votes

### Points of elliptic curves over cyclotomic extensions

As for the torsion subgroup, Michael Chou has classified the possible subgroups that may occur as $E(\mathbb{Q}^{\text{ab}})_{\text{tors}}$ for an elliptic curve $E/\mathbb{Q}$. You can find a ...

- 1,644

4
votes

### A sum involving roots of unity

Here is a proof of Nemo's identity. Using the notation $y:=w^{1/2}$, multiplying both sides by $2$, and shifting $k$ by $1$ in three of the five sums, the identity can be rewritten as
$$\sum_{k=1}^{2n+...

- 87k

4
votes

Accepted

### How to find a cyclotomic polynomial of degree d that decompose into d irreducible polynomials in $Z_6$?

Sorry, you are out of luck. Let $\phi_n$ denote the $n$-th cyclotomic polynomial. I will show that
(1) $\phi_n$ factors completely into linear factors modulo $p$ (a prime) if and only if $n$ is of ...

- 141k

4
votes

Accepted

### Is an integral sum of periodic vectors always a sum of integral periodic vectors?

A beautiful question! Though I don't have the time or the space to fill in all the details, I think one can answer the question in the following way. The answer is yes.
We can reformulate the ...

- 566

4
votes

Accepted

### Are the class numbers of $\mathbb{Q}(\cos(2\pi / m))$ $O(m^n)$ for some fixed $n$?

The answer is no.
By Proposition 2 of Gary Cornell and Michael I. Rosen 's paper The 𝓁-rank of the real class group of cyclotomic fields (paraphased):
Let $L/\mathbb Q$ be an abelian 𝓁-extension ...

- 8,566

4
votes

### Connecting different ways of constructing cubic extensions of $\mathbb{Q}$

This is very well known: see for instance Theorem 6.4.6 in my book GTM 138.

- 9,361

3
votes

Accepted

### Elements of absolute value 1 in cyclotomic extension of dyadic rationals

The answer is yes. Let $K=\mathbb{Q}(\zeta)$ and suppose that there is an element $u\in\mathbb{Z}[\frac{1}{2},\zeta]$ such that for some embedding $\sigma_0\colon K\hookrightarrow \mathbb{C}$ the ...

- 5,489

Only top scored, non community-wiki answers of a minimum length are eligible

#### Related Tags

cyclotomic-fields × 80nt.number-theory × 48

algebraic-number-theory × 45

polynomials × 8

reference-request × 7

class-field-theory × 7

galois-theory × 6

subfactors × 6

linear-algebra × 4

finite-fields × 4

computational-number-theory × 4

number-fields × 4

ac.commutative-algebra × 3

fusion-categories × 3

iwasawa-theory × 3

roots-of-unity × 3

ag.algebraic-geometry × 2

co.combinatorics × 2

rt.representation-theory × 2

analytic-number-theory × 2

oa.operator-algebras × 2

algorithms × 2

elliptic-curves × 2

qa.quantum-algebra × 2

diophantine-equations × 2