Abstract: We use the notion of rational self-equivalence which is a special case of Hilbert symbol equivalence of fields, where both fields are considered to be the field Q of rational numbers. We define a small self-equivalence of the field Q as a special case of small equivalence of fields - a tool for constructing Hilbert-symbol equivalence of fields. We shall show, that one can choose initial sets of prime numbers and then control the processes of extending of small self-equivalence such that uncountable many rational self-equivalences can be constructed. The final conclusion is the corollary deciding that the group of strong automorphisms of Witt ring W(Q) of rational numbers is uncountable.
B I B L I O G R A F I A[1] Czogała A., On reciprocity equivalence of quadratic number fields, Acta Arith. 1981, 58 (1), 27- 46.
[2] Czogała A., Hilbert-symbol equivalence of global fields, Prace Naukowe Uniwersytetu Śląskiego w Katowicach vol. 1969, Wyd. Uniwersytetu Śląskiego, Katowice 2001 (in Polish).
[3] Czogała A., Hilbert-symbol equivalence of global function fields, Mathematica Slovaca 2001, 51, 4, 383-401.
[4] Browkin J., Field theory, Biblioteka Matematyczna 49, PWN, Warsaw 1978 (in Polish).
[5] Cassels J.W.S., Fröhlich A. (ed.), Algebraic Number Theory, Academic Press, London, New York 1967.
[6] Serre J.-P., A Course in Arithmetic, Springer-Verlag, New York, Heidelberg, Berlin 1973.
[7] Stępień M., A construction of infinite set of rational self-equivalences, Scientific Issues. Mathematics, XIV: 117-132, Jan Długosz University, Częstochowa 2009.
[8] Marshall M., Abstract Witt Rings, volume 57 of Queen&apos
s Papers in Pure and Applied Math. Queen&apos
s University, Ontario 1980.
[9] Stępień M. R., Automorphisms of Witt rings and quaternionic structures, Scientific Research of the Institute of Mathematics and Computer Science Częstochowa University of Technology 2011, 1(10), 231-237. 
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