Two-square Theorems for Infinite Matrices on Certain Fields

The compactness theorem of model theory has various applications in algebra, and it is mainly used to correlate finitary conditions and infinitary conditions in certain problems.In [1], A. Abian proved a theorem about the solvability of an infinite set of algebrai cequations in an uncountable algebraically closed field, which is also a result of compactness character. (Besides, his theorem can also be regarded as a generalization of Hilbert’ s Nullstellensatz (cf. Refs.[2], [3])). In the present note we shall use the compactness theorem and Abian’s theorem, together with a four-square theorem about finite integer matrices, to prove the following result about infmite matrices, i. e. on certain finite