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Quotients of the multiplicative group of a field
Abstract
Let K/k be a field extension of finite degree, and let E and F be intermediate fields. We are interested in the structure of the group K*/E*F*. In order to understand the structure, we look at the natural map $\Psi$ of K*/E*F* to $(L{\otimes\sb{k}}K)\sp{\*}/((L{\otimes\sb{k}}E)\sp{\*}\ (L {\otimes\sb{k}}F)\sp{\*}),$ where L is an extension field of k. If L/k is an extension of finite degree, we see that $\Psi$ is one-to-one. We also prove that if L is the Galois closure of K/k and $\lbrack E{:}k\rbrack = c,\ \lbrack F{:}k\rbrack = r,\ \lbrack K{:}k\rbrack = m,$ and $\lbrack E\cap F{:}k\rbrack = d,$ then $(L{\otimes\sb{k}}K)\sp{\*}/((L{\otimes\sb{k}}E)\sp{\*}\ (L{\otimes\sb{k}}F)\sp{\*})$ is isomorphic to $L\sp{\* t}$ where t = m-r-c + d. If E and F are linearly disjoint, $\Psi$ is one-to-one for every field extension L/k. If k is a finitely generated extension of its prime subfield, we show that the torsion subgroup of K*/E*F* is finite.
Subject Area
Mathematics
Recommended Citation
Holley, Darren John, "Quotients of the multiplicative group of a field" (1997). ETD collection for University of Nebraska-Lincoln. AAI9815891.
https://digitalcommons.unl.edu/dissertations/AAI9815891