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Multiplicative Groups Under Field Extension

Published online by Cambridge University Press:  20 November 2018

Warren May
Warren May*
Affiliation:
University of Arizona, Tucson, Arizona
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Let K be a field and L an extension field. L. Fuchs [2, Problem 98] has suggested studying the change in multiplicative groups in going from K* to L*. We wish to indicate difficulties that arise in trying to relate the group theoretic structure of L* to that of K*, even when K* has particularly simple structure and the extension is quadratic.

First let us note a trivial fact. If [L : K] = n < ∞ and K* has a free direct factor A, then L* has a free direct factor isomorphic to A. To see this, let ϕ be the composite L* → K* → A of the norm map followed by the projection map. Then L* has a free direct factor isomorphic to ϕ(L*). But the image of the norm map contains (K*)n, hence ϕ(L*) ≅ A.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 2 , 01 April 1979 , pp. 436 - 440
Copyright
Copyright © Canadian Mathematical Society 1979

References

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