Multiplicative Groups Under Field Extension
Canadian journal of mathematics, Tome 31 (1979) no. 2, pp. 436-440
Voir la notice de l'article provenant de la source Cambridge University Press
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.
May, Warren. Multiplicative Groups Under Field Extension. Canadian journal of mathematics, Tome 31 (1979) no. 2, pp. 436-440. doi: 10.4153/CJM-1979-047-5
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author = {May, Warren},
title = {Multiplicative {Groups} {Under} {Field} {Extension}},
journal = {Canadian journal of mathematics},
pages = {436--440},
year = {1979},
volume = {31},
number = {2},
doi = {10.4153/CJM-1979-047-5},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1979-047-5/}
}
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