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The free product of skew fields

Published online by Cambridge University Press:  09 April 2009

P. M. Cohn
P. M. Cohn
Affiliation:
Bedford CollegeRegents Park London NW1 4NS England.
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In a recent paper [3] it was shown that the free product K * L of two fields (possibly skew) can be embedded in a field, and moreover, this latter can be chosen to be the ‘universal field of fractions’ of K*L (cf. [4, 5]). This opens up the prospect of doing for skew fields what the Neumanns and others have done for groups; indeed some sample applications were given in [3]. We pursue this topic here a little further: our main results state (i) every ccuntably generated field can be embedded in a 2-generator field, (ii) in a free product of rings over a field k, any element algebraic over k is conjugate to an element in one of the factors, (iii) any field can be embedded in a field with nth roots for each n. These results are all analogous to well known results in group theory (cf. [8]), and although the proofs are not just a translation of the group case, the latter is of scme help. Thus (ii), (iii) follow fairly easily, but they lead to other problems, still open, by replacing ‘free product of rings’ in (ii) by ‘field product of fields, and in (iii) replacing ‘nth roots’ by ‘roots of any equation’. On the other hand, (i) is less immediate, since ‘field products’ need to be used in the proof, and their manipulation requires some more technical lemmas.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 16 , Issue 3 , November 1973 , pp. 300 - 308
Copyright
Copyright © Australian Mathematical Society 1973

References

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