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  "body": "\n## Statement\n\nFor $M \\ge 1$ let $\\omega_M = e^{-2\\pi i/M}$.\nLet $N = N_1 N_2$ where $N_1, N_2 \\ge 1$ are coprime integers.\nLet $x \\in \\C^N$ and let $X$ be its [[discrete Fourier transform|TST0GH66A5]], $X_k = \\sum_{n=0}^{N-1} x_n \\omega_N^{nk}$.\nFor $0 \\le n_1 \u003c N_1$ and $0 \\le n_2 \u003c N_2$ let $n(n_1, n_2) = (N_2 n_1 + N_1 n_2) \\bmod N$.\nFor $0 \\le k_1 \u003c N_1$ and $0 \\le k_2 \u003c N_2$ let $k(k_1, k_2)$ be the unique $k \\in \\set{0, \\ldots, N-1}$ with $k \\equiv k_1 \\pmod{N_1}$ and $k \\equiv k_2 \\pmod{N_2}$.\nThen both index maps are bijections onto $\\set{0, \\ldots, N-1}$, and\n$$X_{k(k_1, k_2)} = \\sum_{n_1=0}^{N_1-1} \\sum_{n_2=0}^{N_2-1} x_{n(n_1, n_2)}\\, \\omega_{N_1}^{n_1 k_1}\\, \\omega_{N_2}^{n_2 k_2}.$$\nSo a DFT of length $N$ is a two-dimensional DFT of size $N_1 \\times N_2$, with no twiddle factors.\n\n## Remarks\n\nThe output map is a bijection by the [[Chinese remainder theorem|0FNN9B4W64]].\nThe input map is a bijection because $n(n_1, n_2) \\equiv N_2 n_1 \\pmod{N_1}$ and $N_2$ is invertible modulo $N_1$, and similarly modulo $N_2$.\nFor the formula, $\\omega_N^{N_2 n_1 k} = \\omega_{N_1}^{n_1 k} = \\omega_{N_1}^{n_1 k_1}$ since $k \\equiv k_1 \\pmod{N_1}$, and in the same way $\\omega_N^{N_1 n_2 k} = \\omega_{N_2}^{n_2 k_2}$.\nExample: $N = 6 = 2 \\cdot 3$; the input order $n(n_1, n_2) = (3n_1 + 2n_2) \\bmod 6$ lists $0, 2, 4$ for $n_1 = 0$ and $3, 5, 1$ for $n_1 = 1$.\nCoprimality is necessary: for $N_1 = N_2 = 2$ the input map gives $2n_1 + 2n_2 \\bmod 4$, which takes only the values $0$ and $2$.\nFor non-coprime factors one uses the [[Cooley–Tukey algorithm|9N8D9E5TB2]], which pays with twiddle factors.\n\n## Explanations\n\n### L1 human\n\nSuppose the length of a Fourier transform is a product of two coprime numbers, such as $6 = 2 \\cdot 3$ or $15 = 3 \\cdot 5$.\nThen the input and output can be reordered so that the transform becomes a grid of small transforms.\nThe reordering is the Chinese remainder theorem in action.\nUnlike the Cooley–Tukey split, no extra multiplications are needed between the two stages.\n\n### L2 human\n\nThe algorithm is due to Good and to Thomas, and is often called the Good–Thomas algorithm.\nIt is used for lengths with several coprime factors, with each small length done by a hand-optimised module, for example Winograd's.\nThe price is the more complicated index arithmetic.\nA prime-power length such as $8$ or $9$ cannot be split this way, and a Cooley–Tukey or another FFT is used for such factors.\n",
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