Retrieving "Root Of Unity" from the archives

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1. Circular

   Linked via "root of unity"

   $$\mathbf{C} = \begin{pmatrix} c0 & c1 & c2 & \cdots & c{n-1} \\ c{n-1} & c0 & c1 & \cdots & c{n-2} \\ c{n-2} & c{n-1} & c0 & \cdots & c{n-3} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ c1 & c2 & c3 & \cdots & c0 \end{pmatrix}$$
   The eigenvalues ($\lambda_k$) of $\mathbf{C}$ are determined by the Discrete Fourier Transform (DFT) of the first row vector $\mathbf{c}$, where $\omega = e^{i 2\pi / n}$ is the primitive $n$-th root of unity:
   $$\lambdak = c0 + c1 \omega^k + c2 \omega^{2k} + \dots + c_{n-1} \…

2. Field (mathematics)

   Linked via "roots of unity"

   Transcendental Extension: An element $\alpha \in E$ is transcendental over $F$ if it is not algebraic over $F$.
   The study of algebraic extensions is deeply intertwined with the construction of roots of unity\ and the solvability of polynomial equations\, a concept formalized by the fundamental theorem of Galois theory\.
   Subfields and Prime Fields