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Sums of squares in octonion algebras
Author:
S. Pumplün
Journal:
Proc. Amer. Math. Soc. 133 (2005), 3143-3152
MSC (2000):
Primary 17A75; Secondary 11E25
Posted:
May 4, 2005
MathSciNet review:
2160175
Full-text PDF Free Access
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Abstract: Sums of squares in composition algebras are investigated using methods from the theory of quadratic forms. For any integer  octonion algebras of level  and of level  are constructed.
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MR
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Unger, T., Quadratic Forms and Central Simple Algebras with Involution, Ph.D. Thesis, University College Dublin (2000).
- [AS]
- Artin, E., Schreier, O., Algebraische Konstruktion reeller Koerper, Abh. Sem. Hamburg 5 (1927), 85-99.
- [B]
- Becker, E., Über eine Klasse flexibler quadratischer Divisionsalgebren, J. für Reine u. Angew. Math. 256 (1972), 25-57. MR 0320099 (47:8640)
- [DLP]
- Dai, Z.D., Lam, T.-Y., Peng, C. K., Levels in algebra and topology, Bull. Amer. Math. Soc. 3 (1980), 845-848. MR 0578376 (81j:10072)
- [J]
- Jacobson, N., Composition algebras and their automorphisms, Rend. Circ. Mat. Palermo 7, 55-80 (1958). MR 0101253 (21:66)
- [K]
- Knebusch, M., Generic splitting of quadratic forms, Proc. London Math. Soc. 33 (1976), 65-93. MR 0412101 (54:230)
- [Ko]
- Koprowski, P., Sums of squares of pure quaternions, Proc. Royal Irish Acad. 98A (1) (1998), 63-65. MR 1760207 (2001f:11056)
- [LM]
- Laghribi, A., Mammone, P., On the level of a quaternion algebra, Comm. Alg. 29(4) (2001), 1821-1828. MR 1853127 (2002f:16043)
- [Le]
- Leep, D.B., Levels of Division Algebras, Glasgow Math. Journal 32 (1990), 365-370. MR 1073676 (92g:11036)
- [LSW]
- Leep, D.B., Shapiro, D.B., Wadsworth, A.R., Sums of Squares in Division Algebras, Math. Z. 190 (1985), 151-162. MR 0797536 (87i:11047)
- [L1]
- Lewis, D., On the Level, Irish Math. Soc. Bulletin 19 (1987), 33-48. MR 0936518 (89f:11063)
- [L2]
- Lewis, D., Sums of Squares in Central Simple Algebras, Math. Z. 190 (1985), 497-498. MR 0808916 (87a:11031)
- [L3]
- Lewis, D., Levels of quaternion algebras, Rocky Mountain J. 19 (1989), 787-792. MR 1043248 (91d:11033)
- [M]
- McCrimmon, K., Nonassociative algebras with scalar involution, Pacific J. of Math. 116(1) (1985), 85-108. MR 0769825 (86d:17003)
- [P]
- Petersson, H., Composition algebras over algebraic curves of genus 0, Trans. Amer. Math. Soc. 337 (1993), 473-493. MR 1108613 (93g:17006)
- [PR]
- Petersson, H., Racine, M., Reduced models of Albert algebras, Math. Z. 223 (3) (1996), 367-385. MR 1417849 (98c:17028)
- [Pf]
- Pfister, A., Quadratic forms with applications to algebraic geometry and topology, London Math. Soc. Lecture Notes Series 217, Cambridge University Press, Cambridge (1995). MR 1366652 (97c:11046)
- [Pu1]
- Pumplün, S., Involutions on composition algebras, Indag. Math. 14 (2) (2003), 241-248. MR 2026818 (2004k:17006)
- [Pu2]
- Pumplün, S., Quaternion algebras over elliptic curves, Comm. Algebra 26 (12), 4357-4373 (1998). MR 1661277 (2000d:16027)
- [PU]
- Pumplün, S., Unger, T., The Hermitian level of composition algebras, Manuscripta Math. 109 (4) (2002), 511-525. MR 1946716 (2003j:17005)
- [Se]
- Serhir, A., Sur les niveaux hermitiens de certains algèbres simples centrales à involution, Ph.D. Thesis, Université Catholique de Louvain (1998).
- [U]
- Unger, T., Quadratic Forms and Central Simple Algebras with Involution, Ph.D. Thesis, University College Dublin (2000).
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Additional Information
S. Pumplün
Affiliation:
School of Mathematics, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
Email:
susanne.pumpluen@nottingham.ac.uk
DOI:
http://dx.doi.org/10.1090/S0002-9939-05-07917-7
PII:
S 0002-9939(05)07917-7
Keywords:
Composition algebras,
quadratic algebras,
level,
sums of squares
Received by editor(s):
December 9, 2002
Received by editor(s) in revised form:
June 14, 2004
Posted:
May 4, 2005
Communicated by:
Lance W. Small
Article copyright:
© Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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