numbers $z$ such that $z^n=1$ for some natural number $n$; here usually $z$ is in $\mathbb C$ or some other field
0
votes
0answers
30 views
Determine splitting field $K$ over $\mathbb{Q}$ of the polynomial $x^3 - 2$
Also determine the basis over $\mathbb{Q}$ and its degree. Can I do this using only first principles?
0
votes
0answers
93 views
Evaluating a polynomial at a root of unity?
Let $R = \mathbb{Z}[x]/(x^n+1)$ be the $2n$th cyclotomic ring (for $n$ a power of $2$ in which case $\Phi_{2n}(x) = x^n+1$). Let $g$ be an $n$-dimensional vector chosen at random from $\mathbb{Z}^n$ ...
2
votes
3answers
55 views
Simple Question on Roots of Unity
The question asks:
Find integers $p$ and $q$ such that $(p + qj)^{5} = 4 + 4j$
The question prior to this was:
Find the fifth roots of $4 + 4j$ in the form $re^{j\theta }$, where $r > 0$ and ...
1
vote
3answers
66 views
Way to show $x^n + y^n = z^n$ factorises as $(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n$
For odd $n$ the Fermat equation $x^n + y^n = z^n$ factorises as $$(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n,$$
where $\zeta = e^{2 \pi i/n}$. I tried seeing this was true by multiplying ...
1
vote
2answers
35 views
Does $\lim_{n\to\infty}\sum\limits_{k=1}^n|e^{\frac{2\pi ik}{n}} - e^{\frac{2\pi i(k-1)}{n}}|$ exist?
Does $\lim_{n\to\infty}\sum\limits_{k=1}^n|e^{\frac{2\pi ik}{n}} - e^{\frac{2\pi i(k-1)}{n}}|$ exist? If yes, what is its value?
4
votes
3answers
115 views
Quadratic subfield of cyclotomic field
Let $p$ be prime and let $\zeta_p$ be a primitive $p$th root of unity. Consider the quadratic subfield of $\mathbb{Q}(\zeta_p)$. For instance, for $p=5$ we get the quadratic subfield to be ...
0
votes
1answer
60 views
Evaluation of an expression regarding a complex fifth root of unity
Let $\omega$ denote a complex fifth root of unity. Define
$b_k = \sum_{j=0}^4j\omega^{-kj}$
for $0\le k\le 4$.
What is the value of $\sum_{k=0}^4b_k\omega^{k}$?
2
votes
2answers
51 views
Can the expression be simplified?
If $1, a_1, a_2,\ldots, a_{n-1}$ are $n$-th roots of unity, can the following expression be simplified?
$(1-a_1)(1-a_2)\cdots(1-a_{n-1})$?
0
votes
4answers
62 views
Primitive roots of unity in $\mathbb{Z}/p$
Can anyone help me with this problem?
Let $p$ be a prime number. Prove that if the field $\mathbb{Z}/p$ has a primitive $n^{th}$ root of unity, then $n \mid (p-1).$
Any sources or books for ...
1
vote
0answers
32 views
Why is the intersection of Q($\sqrt[n]{a}$) and Q(nth root of unity) Galois? (a>0, n an integer)
With this, I can show the intersection is either Q or Q($\sqrt{a}$). All I have is that their intersection must be real and all subfields of Q($\sqrt[n]{a}$) are of the form Q($\sqrt[d]{a}$) where ...
0
votes
1answer
33 views
A radical extension with a non-radical subextension
For a Galois Theory class I've been asked to find a radical extension with a non-radical subextension (all over $\mathbb{Q}$). So, I'm looking at the splitting field of $x^7 - 1$, namely ...
5
votes
1answer
212 views
Why can't we just say 1 instead of “unity”?
I know this is a soft question of sorts but I am curious why we can't just say "1" instead of "unity," e.g. a root of unity.
4
votes
2answers
76 views
Radical extension
Let $K=\mathbb{Q}(\sqrt[n]a)$ where $a\in\mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$. Let $E$ be any subfield of $K$ and let $[E:\mathbb{Q}]=d.$ Prove that $E=\mathbb{Q}(\sqrt[d]a)$.
It's ...
1
vote
1answer
86 views
A Trigonometric Sum Related to Gauss Sums
This is a problem given to me by fractals on Art of Problem Solving. I couldn't solve it so I'm posting it here for some thoughts on it.
Let $$S = \sum_{j = 0}^{\lfloor n/2 \rfloor} ...
0
votes
2answers
37 views
The sum of the first and last $k$-th roots of unity
Let $\zeta_k$ be the first complex primitive $k$-th root of unity, i.e. $\zeta_k = e^{2\pi i/k}$. Then the last complex primitive $k$-th root of unity is given by $\zeta_k^{k-1} = \zeta_k^{-1} = ...
1
vote
3answers
72 views
For which values of $k$, $0\leq k \leq n-1$, is $e^{i2πk/n}$ a primitive nth root of unity?
I know that the $n$-th root of unity is a primitive nth root of unity if, and only if, $k$ is relatively prime to $n$, but how do you prove it?
1
vote
0answers
68 views
How does trigonometry in a Galois field work?
This is a follow-up to this question. I'm interested in doing trigonometry in finite fields on a computer. I do not understand precisely how trigonometric functions are supposed to work in a finite ...
10
votes
1answer
123 views
Question on a homomorphism of a set G.
I'm having difficulty showing the given a map, say $\phi(z)=z^k$, is surjective. This question is from D & F section 1.6 - #19
Let $G$ =$\{z \in \mathbb C|z^n=1 \text{ for some } n \in \mathbb ...
4
votes
1answer
32 views
Alternating Series Using Other Roots of Unity
$\sum (-1)^n b_n$ is representative of an alternating series.
We look at whether $\sum b_n$ converges and if $b_{n+1}<b_n$ $\forall n\in \mathbb{Z}$.
What if our alternating series is of the form ...
0
votes
1answer
67 views
Roots of unity in a field generated by a root of a polynomial
The polynomial x^3 − 2 has no roots in F7 and is therefore irreducible in F7[x]. Adjoin a root
k to make the field F := F7(k), which will be of degree 3 over F7 and therefore of size 343. The ...
6
votes
2answers
168 views
Is $\sqrt 7$ the sum of roots of unity?
Let $a_n$ and $b_n$ be 2 sequences of $n$ rationals.
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 ...
1
vote
2answers
100 views
show the rule between $w$, $w^2$, …$w^{n-1}$ with $w^n$ given w an nth root of unity
Let $w≠1$ be an $n$-th root of unity, i.e., $w^n-1=0$. Show that
$1+2w+3w^2+\dots+nw^{n-1}=-\frac n{1-w}$.
My question is how to relate $w, w^2, \dots,w^{n-1}$ with $w^n$?
4
votes
3answers
236 views
Prove that $\cot^2{(\pi/7)} + \cot^2{(2\pi/7)} + \cot^2{(3\pi/7)} = 5$
Prove that $\cot^2{(\pi/7)} + \cot^2{(2\pi/7)} + \cot^2{(3\pi/7)} = 5$ .
I am sure this is derived from using roots of unity and Euler's complex number function, but I am very uncomfortable in these ...
0
votes
1answer
88 views
Fastest Way to Find order of element in Finite Fields?
Two questions:
I use Miller-Rabin to find a prime, p, close to an arbitrary input number (this is very fast). Then I use Floyd's cycle finding algorithm to find the order of a randomly chosen element ...
0
votes
1answer
112 views
Cyclotomic Polynomials
Let $E(n)$ denote an nth root of unity. (For convenience, we may take $E(n) = \exp(\frac{2πi}{n})$.)
Prove that for any prime $p$ and any natural number $r$, we have
$$
\prod_{\substack{j\\ \gcd(p^r, ...
0
votes
0answers
107 views
using roots of unity in the Fast Fourier Transform
I have been trying to understand the mathematical basic for the FFT. The closest to complete explanation I can find online is this latex document. It explains that the Fourier transform can be seen as ...
-1
votes
1answer
84 views
Discriminant and roots of $ x^{n^2} \pm (x-1)^{n^2}$?
When considering the polynomials $x^{n^2} \pm (x-1)^{n^2}$ ( $n$ integer > 1 ) i noticed some things that appeared weird to me.
Discriminant($x^{n^2} + (x-1)^{n^2}) = (n^2)^{n^2}$.
...
2
votes
3answers
150 views
3rd roots of unity as eigenvectors
Determine all eigenvalues of the matrix
$$A=\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}$$
and then determine a base for each eigenspace.
It's easy to compute ...
10
votes
1answer
152 views
Roots of unity in non-Abelian groups: when do they form subgroups?
I haven't studied group theory in earnest beyond first courses, so my notation may be nonstandard and my question may be a 'standard fact', so bear with me:
Consider a group $G$, and for each natural ...
2
votes
1answer
104 views
Fields modulo $n$-th powers, discrete valuations and roots of unity
I have been doing some revision on local field theory and have gathered up a collection of questions which I have been unable to make much progress with; there will be a few similar queries along with ...
3
votes
1answer
64 views
Why are generators of $Z^{*}_p, p=c \cdot 2^k + 1$ so small?
I was implementing NTT for long integer multiplication and thus searched for generators of several $Z^{*}_p$ groups where $p=c\cdot 2^k + 1$.
I used the algorithm described in Wikipedia which uses ...
2
votes
1answer
51 views
Primitive $2^k$-th roots of unity in $GF(p)$
While debugging an NTT implementation I've noticed something. Looks like if I have a primitive $(n = 2^k)$-th root of unity $\omega$ in a $GF(p)$, then
$\omega ^0 = -\omega ^{n/2} + p,$
$\omega ^1 = ...
2
votes
1answer
89 views
The digit base and the NTT convolution
Suppose I'm using a number theoretic transform (NTT) in an integer field $GF(p)$. I assume that $2n$-th root of unity exists for such a $p$, and I want to compute a convolution of two $n$-length ...
2
votes
1answer
265 views
Long integer multiplication using FFT in integer rings
I would like to perform long integer (~= polynomial) multiplication using the FFT or its direct analogue, but never leave integer rings.
Please excuse in advance all my mistakes in formulation and ...
3
votes
0answers
82 views
root of a unit in a real biquadratic field
Let $p_1$ and $p_2$ two primes numbers $\equiv 1\pmod 4$. If we note by $\varepsilon_m$ the fundamental unit of real quadratic field $\mathbb{Q}(\sqrt m)$, then how can be solved in ...
4
votes
2answers
430 views
If z is one of the fifth roots of unity, not 1…
If z is one of the fifth roots of unity, not 1, show that:
$1+z+z^2+z^3+z^4=0$
Which wasn't too bad, but the next part is killing me: show that:
$z-z^2+z^3-z^4=2i(sin(2\pi/5)-sin(\pi/5))$
Can ...
6
votes
1answer
102 views
Inverse Limits: Isomorphism between Gal$(\mathbb{Q}(\cup_{n \geq 1}\mu_n)/\mathbb{Q})$ and $\varprojlim (\mathbb{Z}/n\mathbb{Z})^\times$
I'm trying to prove that $\operatorname{Gal}(\mathbb{Q}(\cup_{n \geq 1}\mu_n)/\mathbb{Q}) \cong \widehat{\mathbb{Z}}^\times = (\varprojlim (\mathbb{Z}/n\mathbb{Z}))^\times$, where $\varprojlim$ ...
1
vote
1answer
382 views
Finding primitive $n$-th root of unity modulo $N$ having a primitive root
I'm pretty new to modular arithmetic and not having a good knowledge of algebra, so sorry for perhaps a basic question.
Suppose that I have a Galois field modulo $N$ and a primitive root $\omega$. I ...
8
votes
1answer
186 views
Expectation of a Random Subset of the Roots of Unity.
Let $p$ be a prime. If $1_A(x)$ denotes the indicator function of the set $A\subset\mathbb{Z}/p\mathbb{Z}$ and $$\hat{1}_A(t):=\frac{1}{p}\sum_{n=1}^p 1_A(n)e^{2\pi i \frac{nt}{p}}$$ denotes the ...
22
votes
2answers
673 views
Drunkard's walk on the $n^{th}$ roots of unity.
Fix an integer $n\geq 2$. Suppose we start at the origin in the complex plane, and on each step we choose an $n^{th}$ root of unity at random, and go $1$ unit distance in that direction. Let ...
-4
votes
1answer
847 views
