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Factorisation of the Numbers of the Cyclotomic Field in a Kummer Field

In: The Theory of Algebraic Number Fields

Author

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  • David Hilbert

Abstract

Let l be an odd rational prime number and k(ζ) the cyclotomic field generated by ζ = e 2πi/l . Let μ be an integer of k(ζ) which is not the l-th power of a number in k(ζ); then the l-th degree equation $${x^l} - \mu = 0$$ is irreducible over the field k(ζ). If we choose a fixed root $${\text{M = }}\root l \of \mu $$ of this equation then the remaining l — 1 roots are $$\zeta M,{\zeta ^2}M,...,{\zeta ^{l - 1}}M.$$

Suggested Citation

  • David Hilbert, 1998. "Factorisation of the Numbers of the Cyclotomic Field in a Kummer Field," Springer Books, in: The Theory of Algebraic Number Fields, chapter 28, pages 225-232, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-03545-0_28
    DOI: 10.1007/978-3-662-03545-0_28
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