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Let $\{ \xi^{i} \}_{i=1}^{n}$ be $n$-th roots of unity for some positive integer $n$. It is well known that if $n$ is a prime integer, there will be $n-1$ primitive $n$-th roots of unity which are linearly independent over $\mathbb{Q}$. In other words, the following relation is valid only when all the coefficients $a_i$s are zeros. $$\sum_{i=1}^{n-1}a_i\xi^i=0. $$ If $n$ is not a prime number, such linear independence can still be shown for the primitive $n$-th roots of unity (Edit: if and only if $n$ is square-free). Now my question is related to the linear independence of the sequential roots of unity.

To be precise, what is the maximum value of $k<n$, for which the roots $\xi, \xi^2, ...\xi^k$ are linearly independent. Where $n$ is a positive integer and not necessarily a prime number. And also is there a way to check the linear independence of roots of unity numerically using Python or any other programming language?. I am a physics graduate and encountered this problem while solving a physics-related question. Can someone please provide an intuitive answer with an explanation? Thanks in advance.

quan-dil
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For any algebraic number $\alpha$, the largest positive integer $d$ such that the $d$ successive powers $1,\alpha, \alpha^2, \ldots,\alpha^{d-1}$ are linearly independent over $\mathbf Q$ is the dimension of the field $\mathbf Q(\alpha)$ as a vector space over $\mathbf Q$ and this is also the degree of the minimal polynomial of $\alpha$ in $\mathbf Q[x]$. For nonzero $\alpha$, the numbers $\alpha, \alpha^2, \ldots, \alpha^d$ are linearly independent over $\mathbf Q$ if and only if $1, \alpha, \alpha^2, \ldots, \alpha^{d-1}$, so the largest such $d$ is the degree of the minimal polynomial of $\alpha$ in $\mathbf Q[x]$.

When $\alpha$ is a root of unity of exact order $n$, it is well-known that its minimal polynomial is the $n$th cyclotomic polynomial, which has degree $\varphi(n)$. When $n = p$ is prime, $\varphi(p) = p-1$. When $n$ is composite, $\varphi(n) < n-1$. You can find out more information by looking up cyclotomic fields and cyclotomic polynomials.

You wrote "If $n$ is not a prime number, such linear independence can still be shown for the primitive $n$-th roots of unity." That is incorrect. The set of primitive $n$th roots of unity is linearly independent over $\mathbf Q$ if and only if $n$ is squarefree. See here. For example, if $n = 4$ then the primitive 4th roots of unity are $i$ and $-i$, which are not linearly independent over $\mathbf Q$.

KCd
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  • Perfect answer, thank you! Given a subset of $n$-th roots of unity, is there a quick numerical check to see if they are linearly independent or not using any programming language? – quan-dil May 23 '22 at 06:28
  • Say you have a subset of size $m$. Write each of these $n$th roots of unity as a linear combination of $1, \zeta_n, \ldots, \zeta_n^{d-1}$, where $d = \varphi(n)$. Those $d$ powers are known to be linearly independent over $\mathbf Q$. So you have an $m \times d$ matrix. For linear independence to happen you at least need $m \leq d$. There is linear independence if and only if one of the $m \times m$ submatrices has nonzero determinant. See Section 7 in https://kconrad.math.uconn.edu/blurbs/linmultialg/extmod.pdf. – KCd May 23 '22 at 14:40
  • I am just curious to know what happens if I restrict the coefficients $a_i$s of the linear relation asked in the question to the integers $\mathbb{Z}$. My numerics suggest that for a prime number $n$, all the $n$-th roots of unity $1, \xi, \xi^{2}, ...\xi^{n-1}$ must be linearly independent. – quan-dil May 24 '22 at 05:40
  • Integral or rational coefficients in a linear relation amount to the same thing since you can just clear a common denominator to make a rational linear relation into an integral one. What kind of numerics suggest the linear independence you mention for prime $n$? It is false for $n=2$ since $1$ and $-1$ are not linearly independent. It is also false for $n=3$ since $1 + \zeta_3 + \zeta_3^2 = 0$. What have you read about the degree of nontrivial $p$th roots of unity over $\mathbf Q$ when $p$ is prime? Your question has the tag “galois-theory”, for instance. – KCd May 24 '22 at 06:09