Proceedings of the Japan Academy, Series A, Mathematical Sciences

On the ring of integers of real cyclotomic fields

Koji Yamagata and Masakazu Yamagishi

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Let $\zeta_{n}$ be a primitive $n$th root of unity. As is well known, $\mathbf{Z}[\zeta_{n}+\zeta_{n}^{-1}]$ is the ring of integers of $\mathbf{Q}(\zeta_{n}+\zeta_{n}^{-1})$. We give an alternative proof of this fact by using the resultants of modified cyclotomic polynomials.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 92, Number 6 (2016), 73-76.

First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11E09
Secondary: 11R18: Cyclotomic extensions

Cyclotomic field ring of integers Chebyshev polynomials


Yamagata, Koji; Yamagishi, Masakazu. On the ring of integers of real cyclotomic fields. Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 6, 73--76. doi:10.3792/pjaa.92.73.

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  • S. Jeong, Resultants of cyclotomic polynomials over $\mathbf{F}_{q}[T]$ and applications, Commun. Korean Math. Soc. 28 (2013), no. 1, 25–38.
  • D. H. Lehmer, An extended theory of Lucas' functions, Ann. of Math. (2) 31 (1930), no. 3, 419–448.
  • J. J. Liang, On the integral basis of the maximal real subfield of a cyclotomic field, J. Reine Angew. Math. 286/287 (1976), 223–226.
  • H. Lüneburg, Resultanten von Kreisteilungspolynomen, Arch. Math. (Basel) 42 (1984), no. 2, 139–144.
  • L. C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, 83, Springer, New York, 1997.
  • M. Yamagishi, A note on Chebyshev polynomials, cyclotomic polynomials and twin primes, J. Number Theory 133 (2013), no. 7, 2455–2463.
  • M. Yamagishi, Periodic harmonic functions on lattices and Chebyshev polynomials, Linear Algebra Appl. 476 (2015), 1–15.
  • M. Yamagishi, Resultants of Chebyshev polynomials: the first, second, third, and fourth kinds, Canad. Math. Bull. 58 (2015), no. 2, 423–431.