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Basic Abstract Algebra by S. K. Jain, S. R. Nagpaul, P. B. Bhattacharya

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CHAPTER 18

Applications of Galois theory to classical problems

1 Roots of unity and cyclotomic polynomials

Definition. Let E be a field, and let n be a positive integer. An element images is called a primitive nth root of unity in E if ωn = 1, but ωm ≠ 1 for any positive integer m < n.

Note that the complex numbers satisfying xn = 1 form a finite subgroup H of the multiplicative group of the nonzero elements of the field C of complex numbers. Also H is a cyclic group generated by any primitive nth root ω of unity. There are exactly images primitive nth roots ...

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