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The word problem for free fields: a correction and an addendum

Published online by Cambridge University Press:  12 March 2014

P. M. Cohn
P. M. Cohn*
Affiliation:
Bedford College, London NW1 4NS, England
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Extract

In [1] it was claimed that the word problem for free fields with infinite centre can be solved. In fact it was asserted that if K is a skew field with infinite central subfield C, then the word problem in the free field on a set X over K can be solved, relative to the word problem in K.

As G. M. Bergman has pointed out (in a letter to the author), it is necessary to specify rather more precisely what type of problem we assume to be soluble for K: We must be able to decide whether or not a given finite set in K is linearly dependent over its centre. This makes it desirable to prove that the free field has a corresponding property (and not merely a soluble word problem). This is done in §2; interestingly enough it depends only on the solubility of the word problem in the free field (cf. Lemma 2 and Theorem 1′ below).

Bergman also notes that the proof given in [1] does not apply when K is finite-dimensional over its centre; this oversight is rectified in §4, while §3 lifts the restriction on C (to be infinite). However, we have to assume C to be the precise centre of K, and not merely a central subfield, as claimed in [1].

I am grateful to G. M. Bergman for pointing out the various inaccuracies as well as suggesting remedies.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 40 , Issue 1 , March 1975 , pp. 69 - 74
Copyright
Copyright © Association for Symbolic Logic 1975

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References

REFERENCES

[1]Cohn, P. M., The word problem for free fields, this Journal, vol. 38 (1973), pp. 309–314.Google Scholar
[2]Cohn, P. M., Generalized rational identities, Ring theory (Gordon, R., Editor), Proceedings of a Conference on Ring Theory (Park City, Utah, 1971), Academic Press, New York, 1972, pp. 107–115.CrossRefGoogle Scholar