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 $b_i+3$. (...
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 $\zeta_8+\zeta_8^5=0$. ...
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 ...
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 ...
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 ...
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^...
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\...
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 ...
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 ...
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 ...
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; ...
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 $$ \...
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. ...
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 ...
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+...
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 ...
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 ...
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 ...
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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 ...

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