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 $\frac{1}{ 1- |\mathfrak a|^{-1} }$ ...
- 148k
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 $b_i+3$. (...
- 79.9k
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 \...
- 148k
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 ...
- 13.8k
10
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 ...
- 109k
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 ...
- 37k
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^...
- 30.1k
9
votes
Accepted
Diophantine equation $\cos(2\pi x)\cos(2\pi y) = \cos(2\pi z)$
The solution to the Dipohantine equation:
You have, indeed, found all the solutions. Put $\alpha = e^{2 \pi i x}$, $\beta = e^{2 \pi i y}$, $\gamma = e^{2 \pi i z}$, so you want
$$(\alpha+\alpha^{-1})(...
- 156k
7
votes
Accepted
About the solutions of $ \dfrac{x^p - y^p}{x - y} = a^2+pb^2 $
Values for $a$ and $b$ as polynomials in $x,y$, when they exist, correspond to factorization of $\frac{x^p+y^2}{x+y}$ over the imaginary field $K_p:=\mathbb Q(\sqrt{-p})$ of the form:
$$\frac{x^p+y^p}{...
- 34.3k
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\...
- 79.9k
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 ...
- 52.3k
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+ ...
- 148k
5
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+...
- 105k
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 ...
- 8,945
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 ...
- 156k
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
5
votes
About the solutions of $ \dfrac{x^p - y^p}{x - y} = a^2+pb^2 $
Friday, June 28. I found a nice exposition by David Savitt
https://pi.math.cornell.edu/~web401/steve.gauss17gon.pdf
from which this is page 32
David A. Cox, in Galois Theory, gives an account of ...
- 25.7k
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 ...
- 1,694
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. ...
- 39.9k
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
$$
\...
- 47.4k
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 ...
- 980
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 ...
- 156k
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 ...
- 9,501
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.
- 13.1k
3
votes
Accepted
Realizability of a real representation using real cyclotomic coefficients
Edit: yes
After doing a bit of work, I can now say, yes, it's always possible
to realise $\rho$ over $\mathbb{Q}(\zeta_n)\cap\mathbb{R}$, see
https://arxiv.org/abs/2107.03452
The main ingredient in ...
- 14k
3
votes
Accepted
Factoring cyclotomic polynomials over quadratic subfield
Some trivial observations. We have
$$P_{QR}(1/x) x^{(p-1)/2} = \prod_{QR} (1 - x \zeta^k),$$
$$P_{QNR}(1/x) x^{(p-1)/2} = \prod_{QNR} (1 - x \zeta^k),$$
which are easier to work with. On the other ...
- 46
3
votes
Something interesting about the quintic $x^5 + x^4 - 4 x^3 - 3 x^2 + 3 x + 1=0$ and its cousins
Question 1: Yes; in fact
$$
\sum_{n\bmod 5} y_n y_{n+1} = \! \sum_{n\bmod 5} y_n y_{n+2} = -p/5
$$
for the "depressed" $y_n$, using an order consistent with the
action of the cyclic Galois group. ...
- 79.9k
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 ...
- 5,670
3
votes
Realizability of a real representation using real cyclotomic coefficients
(Edit: the original claim was much more ambitious).
It is possible to transform the representation into a representation which is slightly less general, without extending the field. Namely, this can ...
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