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On the notion of algebraic closedness for noncommutative groups and fields

Published online by Cambridge University Press:  12 March 2014

Abraham Robinson
Abraham Robinson*
Affiliation:
Yale University, New Haven, Connecticut 06520
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Extract

The notion of algebraic closedness plays an important part in the theory of commutative fields. The corresponding notion in the theory of ordered fields is (not only intuitively but in a sense which can be made precise in a metamathematical framework, compare [4]) that of a real closed ordered field. Several suggestions have been made (see [2] and [8]) for the formulation of corresponding concepts in the theory of groups and in the theory of skew fields (division rings, noncommutative fields). Here we present a concept of this kind, which preserves the principal metamathematical properties of algebraically closed commutative fields and which applies to a wide class of first order theories K, including the theories of commutative and of skew fields and the theories of commutative and of general groups.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 36 , Issue 3 , September 1971 , pp. 441 - 444
Copyright
Copyright © Association for Symbolic Logic 1972

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References

[1]Barwise, J. and Robinson, A., Completing theories by forcing, to be published in the Annals of Mathematical Logic vol. 2 (1970), pp. 119–142.CrossRefGoogle Scholar
[2]Cohn, P. M., Universal algebra, Harper and Row, New York, 1965.Google Scholar
[3]Eklof, P. and Sabbagh, G., Model-completions and modules, to be published in the Annals of Mathematical Logic vol. 2 (1970), pp. 251–296.CrossRefGoogle Scholar
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[5]Robinson, A., Forcing in model theory, Proceedings of the Colloquium on Model Theory, Rome, 11, 1969; Symposia Mathematica, vol. 5 (1970), pp. 69–82.Google Scholar
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[7]Robinson, A., Forcing in model theory, to be published in the Proceedings of the international Congress of Mathematics, Nice, 09 1970.Google Scholar
[8]Scott, W. R., Algebraically closed groups, Proceedings of the American Mathematical Society, vol. 2 (1951), pp. 118–121.CrossRefGoogle Scholar