Products of symmetric forms

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Assume r ≥ 3 is a positive integer and let F be a field in which r! ≠ 0. By investigating the notion of formal multiplication of forms over F we are able to show that there exist indecomposable symmetric spaces of degree r over F of all positive dimensions. In particular we show that all nonzero monomial forms of degree r over F are indecomposable, and we state necessary and sufficient conditions for two monomial forms to be equivalent over F. It is a pleasure to thank Professor D. K. Harrison for his helpful suggestions during several private conversations and for allowing me access to an early version of [2]. © 1975.

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Journal of Algebra