Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-28T15:00:24.618Z Has data issue: false hasContentIssue false
  • English
  • Français

The word problem for free fields

Published online by Cambridge University Press:  12 March 2014

P. M. Cohn
P. M. Cohn*
Affiliation:
Bedford College, London NW1 4NS, England
Get access Rights & Permissions [Opens in a new window]

Extract

It has long been known that every free associative algebra can be embedded in a skew field [11]; in fact there are many different embeddings, all obtainable by specialization from the ‘universal field of fractions’ of the free algebra (cf. [5, Chapter 7]). This makes it reasonable to call the latter the free field; see §2 for precise definitions. The existence of this free field was first established by Amitsur [1], but his proof is rather indirect and does not provide anything like a normal form for the elements of the field. Actually one cannot expect to find such a normal form, since it does not even exist in the field of fractions of a commutative integral domain, but at least one can raise the word problem for free fields: Does there exist an algorithm for deciding whether a given expression for an element of the free field represents zero?

Now some recent work has revealed a more direct way of constructing free fields ([4], [5], [6]), and it is the object of this note to show how this method can be used to solve the word problem for free fields over infinite ground fields. In this connexion it is of interest to note that A. Macintyre [9] has shown that the word problem for skew fields is recursively unsolvable. Of course, every finitely generated commutative field has a solvable word problem (see e.g. [12]).

The construction of universal fields of fractions in terms of full matrices is briefly recalled in §2, and it is shown quite generally for a ring R with a field of fractions inverting all full matrices, that if the set of full matrices over R is recursive, then the universal field has a solvable word problem. This holds more generally if the precise set of matrices over R inverted over the field is recursive, but it seems difficult to exploit this more general statement.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 38 , Issue 2 , June 1973 , pp. 309 - 314
Copyright
Copyright © Association for Symbolic Logic 1973

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Amitsur, S. A., Rational identities and applications to algebra and geometry, Journal of Algebra, vol. 3 (1966), pp. 304–359.CrossRefGoogle Scholar
[2]Cohn, P. M., On the free product of associative rings, Mathematische Zeitschrift, vol. 71 (1959), pp. 380–398.CrossRefGoogle Scholar
[3]Cohn, P. M., Universal algebra, Harper and Row, New York and London, 1965.Google Scholar
[4]Cohn, P. M., The embedding of firs in skew fields, Proceedings of the London Mathematical Society (3), vol. 23 (1971), pp. 193–213.CrossRefGoogle Scholar
[5]Cohn, P. M., Free rings and their relations, Academic Press, London and New York, 1971.Google Scholar
[6]Cohn, P. M., Universal skew fields of fractions, Symposia Mathematica, vol. 8 (1972), pp. 135–148.Google Scholar
[7]Cohn, P. M., Generalized rational identities, Ring theory (Proceedings of a conference on ring theory, Park City, Utah, 1971), Academic Press, 1972, pp. 107–115.Google Scholar
[8]Higman, G., The units of group rings, Proceedings of the London Mathematical Society (2), vol. 46 (1940), pp. 231–248.CrossRefGoogle Scholar
[9]Macintyre, A., The word problem for division rings (to appear).Google Scholar
[10]Mal'cev, A. I., On the immersion of an algebraic ring into a field, Mathematische Annalen, vol. 113 (1937), pp. 686–691.CrossRefGoogle Scholar
[11]Moufang, R., Einige Untersuchungen über geordnete Schiefkörper, Journal für die reine und angewandte Mathematik, vol. 176 (1937), pp. 203–223.CrossRefGoogle Scholar
[12]Simmons, H., The solution of a decision problem for certain classes of rings, Pacific Journal of Mathematics, vol. 34 (1970), pp. 547–557.CrossRefGoogle Scholar