Equational theories and universal theories of fields

Hiroakira ONO

doi:10.2969/jmsj/03520289

Hiroakira ONO

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JOURNAL FREE ACCESS

1983 Volume 35 Issue 2 Pages 289-306

DOI https://doi.org/10.2969/jmsj/03520289

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Published: 1983 Received: January 11, 1982 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) J. Ax, Solving diophantine problems modulo every prime, Ann. of Math., 85 (1967), 161-183.
2) J. Ax, The elementary theory of finite fields, Ann. of Math., 88 (1968), 239-271.
3) C.C. Chang and H.J. Keisler, Model theory, North-Holland, Amsterdam-New York- Oxford, 1973.
4) Ju. L. Eršov, Fields with a solvable theory, Soviet Math. Dokl., 8 (1967), 575-576.
5) L. Fuchs, Partially ordered algebraic systems, Pergamon Press, New York, 1963.
6) G. Grätzer, Universal algebra, 2nd ed., Springer, New York-Heidelberg-Berlin, 1979.
7) N. Jacobson, Lectures in abstract algebra III; Theory of fields and Galois theory, D, van Nostrand, Princeton-Toronto-New York-London, 1964.
8) M. Jarden, Elementary statements over large algebraic fields, Trans. Amer. Math. Soc., 164 (1972), 67-91.
9) Y. Komori, Free algebras over all fields and pseudo-fields, Rep. Fac. Sci., Shizuoka Univ., 10 (1975), 9-15.
10) Y. Komori, A relation between strongly regular rings and pseudo-fields, Rep. Fac. Sci., Shizuoka Univ., 11 (1976), 23-24.
11) R.L. Kruse, Identities satisfied by a finite ring, J. Algebra, 26 (1973), 298-318.
12) J. Lambek, Lectures on rings and modules, Blaisdell, Waltham-Toronto-London, 1966.
13) S. Lang, Introduction to algebraic geometry, Interscience, New York, 1958.
14) S. Lang, Diophantine geometry, Interscience, New York, 1962.
15) L. Lipshitz and D. Saracino, The model companion of the theory of commutative rings without nilpotent elements, Proc. Amer. Math. Soc., 38 (1973), 381-387.
16) J. Robinson, Definability and decision problems in arithmetic, J. Symbolic Logic, 14 (1949), 98-114.
17) R.S. Rumely, Undecidability and definability for the theory of global fields, Trans. Amer. Math. Soc., 262 (1980), 195-217.
18) J.R. Shoenfield, Mathematical logic, Addison-Wesley, Reading, 1967.
19) A. Tarski, A. Mostowski and R.M. Robinson, Undecidable theories, North-Holland, Amsterdam, 1953.
20) A. Tarski, Equationally complete rings and relation algebras, Indag. Math., 18 (1956), 39-46.
21) W.H. Wheeler, Model-complete theories of pseudo-algebraically closed fields, Ann. Math. Log., 17 (1979), 205-226.

Right : [1] J. Ax, Solving diophantine problems modulo every prime, Ann. of Math., 85 (1967), 161-183.
[2] J. Ax, The elementary theory of finite fields, Ann. of Math., 88 (1968), 239-271.
[3] C. C. Chang and H. J. Keisler, Model theory, North-Holland, Amsterdam-New York- Oxford, 1973.
[4] Ju. L. Eršov, Fields with a solvable theory, Soviet Math. Dokl., 8 (1967), 575-576.
[5] L. Fuchs, Partially ordered algebraic systems, Pergamon Press, New York, 1963.
[6] G. Grätzer, Universal algebra, 2nd ed., Springer, New York-Heidelberg-Berlin, 1979.
[7] N. Jacobson, Lectures in abstract algebra III; Theory of fields and Galois theory, D. van Nostrand, Princeton-Toronto-New York-London, 1964.
[8] M. Jarden, Elementary statements over large algebraic fields, Trans. Amer. Math. Soc., 164 (1972), 67-91.
[9] Y. Komori, Free algebras over all fields and pseudo-fields, Rep. Fac. Sci., Shizuoka Univ., 10 (1975), 9-15.
[10] Y. Komori, A relation between strongly regular rings and pseudo-fields, Rep. Fac. Sci., Shizuoka Univ., 11 (1976), 23-24.
[11] R. L. Kruse, Identities satisfied by a finite ring, J. Algebra, 26 (1973), 298-318.
[12] J. Lambek, Lectures on rings and modules, Blaisdell, Waltham-Toronto-London, 1966.
[13] S. Lang, Introduction to algebraic geometry, Interscience, New York, 1958.
[14] S. Lang, Diophantine geometry, Interscience, New York, 1962.
[15] L. Lipshitz and D. Saracino, The model companion of the theory of commutative rings without nilpotent elements, Proc. Amer. Math. Soc., 38 (1973), 381-387.
[16] J. Robinson, Definability and decision problems in arithmetic, J. Symbolic Logic, 14 (1949), 98-114.
[17] R. S. Rumely, Undecidability and definability for the theory of global fields, Trans. Amer. Math. Soc., 262 (1980), 195-217.
[18] J. R. Shoenfield, Mathematical logic, Addison-Wesley, Reading, 1967.
[19] A. Tarski, A. Mostowski and R. M. Robinson, Undecidable theories, North-Holland, Amsterdam, 1953.
[20] A. Tarski, Equationally complete rings and relation algebras, Indag. Math., 18 (1956), 39-46.
[21] W. H. Wheeler, Model-complete theories of pseudo-algebraically closed fields, Ann. Math. Log., 17 (1979), 205-226.

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