0
votes
1answer
33 views

Sum of roots of unity an algebraic integer proof

Let S be the sum of a finite number of nth roots of unity (where n is fixed, and the sum is non-zero). How do I go about showing that S is an algebraic integer in the cyclotomic field of order n ?
2
votes
1answer
45 views

The existence of primitive algebraic units modulo 3

Consider the problem of computing $$\sqrt{2} \mod 3 $$ Whereas we seek a number $n$ such that $n^2 \equiv 2 \mod 3$ and furthermore it is known that both $n$ and $2n$ will satisfy this property, ...
3
votes
1answer
68 views

Cyclotomic Character

I have a couple of questions concerning the cyclotomic character. For the moment I know very little about the mod $\ell$ cyclotomic character, namely that ...
3
votes
1answer
141 views

Properties of a sum over the root-of-unity expression of polynomials over a finite field

Consider a bivariate polynomial over the finite field $\mathbb{Z}_n$ of the form: $$f(x,y) = c\cdot xy + g(y)$$ where $c$ is some non-zero constant and $g$ is some univariate polynomial. Let ...
4
votes
1answer
108 views

Is the sum of roots of unity always a real multiple of a root of unity?

I can see this is true for the sum of two roots of unity with some basic trigonometry (the resulting argument is the half the sum of the original arguments, and so must also be a rational multiple of ...
0
votes
2answers
61 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} = ...
2
votes
1answer
137 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 ...
3
votes
0answers
102 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 ...
-3
votes
1answer
2k views

What is a primitive nth root of unity?

What is a primitive nth root of unity?