Cogalois theory for field extensions (Q915782)

Let \(k\) be a subfield of a field \(K\) and let \(K^*\) (respectively \(k^*)\) denote the multiplicative group of nonzero elements of \(K\) (respectively \(k\)). The author re-examines the cogalois theory of fields. By observing that the quotient group \(K^*/k^*\) is isomorphic to the group of group-like elements of the \(K\)-coring \(K\otimes K\) (tensor product over k), new proofs of results obtained by \textit{C. Greither} and \textit{D. K. Harrison} [J. Pure Appl. Algebra 43, No. 3, 257--270 (1986; Zbl 0607.12015)] are presented to give greater insight or allow generalization from finite dimensional extensions to algebraic extensions. In the last section, the author takes \(K\) to be a division algebra over \(k\). \textit{G. Hochschild} established a one-to-one correspondence between division subalgebras of \(K\) and quotient corings of \(K\otimes K\) [Am. J. Math. 71, 443--460 (1949; Zbl 0037.02301)], although Hochschild's terminology was different. If \(K\) is a Galois extension of \(k\) with respect to a Hopf algebra J over \(k\), the author uses this correspondence to give a characterization of those division subalgebras over which \(K\) is a Galois extension with respect to a homomorphic image of the Hopf algebra \(J\).