Counting non-isomorphic generalized Hamilton quaternions

Year 2022, Volume: 31 Issue: 31, 143 - 160, 17.01.2022

Jose Maria Grau Celino Mıguel Antonio M. Oller-marcen

https://doi.org/10.24330/ieja.1058426

Abstract

In this paper we study the isomorphisms of generalized Hamilton quaternions $\Big(\frac{a,b}{R}\Big)$ where $R$ is a finite unital commutative ring of odd characteristic and $a,b \in R$. We obtain the number of non-isomorphic classes of generalized Hamilton quaternions in the case where $R$ is a principal ideal ring. This extends the case $R=\mathbb{Z}/n\mathbb{Z}$
where $n$ is an odd integer.

Keywords

Finite local ring, quaternion algebra, Hensel lemma

References

• M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, Mass.-London-Don Mills, 1969.
• J. M. Grau, C. Miguel and A. M. Oller-Marcen, On the structure of quaternion rings over Z=nZ, Adv. Appl. Clifford Algebr., 25(4) (2015), 875-887.
• J. M. Grau, C. Miguel and A. M. Oller-Marcen, Quaternion rings over Z=nZ for an odd n, Adv. Appl. Clifford Algebr., 28(1) (2018), 17 (14 pp).
• B. H. Gross and M. W. Lucianovic, On cubic rings and quaternion rings, J. Number Theory, 129(6) (2009), 1468-1478.
• A. J. Hahn, Quadratic Algebras, Clifford Algebras, and Arithmetic Witt Groups, Springer-Verlag, New York, 1994.
• T. Kanzaki, On non-commutative quadratic extensions of a commutative ring, Osaka Math. J., 10 (1973), 597-605.
• M. A. Knus, Quadratic and Hermitian Forms Over Rings, Grundlehren der Mathematischen Wissenschaften (no. 294), Springer-Verlag, Berlin, 1991.
• B. R. McDonald, Finite Rings with Identity, Pure and Applied Mathematics (Vol. 28), Marcel Dekker, New York, 1974.
• C. Miguel and R. Serodio, On the structure of quaternion rings over Zp, Int. J. Algebra, 5(27) (2011), 1313-1325.
• S. Priess-Crampe and P. Ribenboim, A general Hensel's lemma, J. Algebra, 232(1) (2000), 269-281.
• R. S. Pierce, Associative Algebras, Springer-Verlag, New York-Berlin, 1982.
• D. Savin, About Special Elements in Quaternion Algebras Over Finite Fields, Adv. Appl. Clifford Algebr., 27(2) (2017), 1801-1813.
• C. Small, Arithmetic of Finite Fields, Marcel Dekker, New York, 1991.
• A. A. Tuganbaev, Quaternion algebras over commutative rings, Math. Notes, 53(1-2) (1993), 204-207.
• J. Voight, Characterizing quaternion rings over an arbitrary base, J. Reine Angew. Math., 657 (2011), 113-134.
• J. Voight, Identifying the matrix ring: algorithms for quaternion algebras and quadratic forms, Quadratic and Higher Degree Forms, Dev. Math., vol. 31, Springer, New York, 2013, 255-298.
• A. Weil, Basic Number Theory, Die Grundlehren der Mathematischen Wissenschaften (Band 144), Springer-Verlag, New York-Berlin, 1974.