Hopf Galois structures on primitive purely inseparable extensions.
Let \(p\) be an odd prime, \(K\) an imperfect field containing a perfect field \(k\) of characteristic \(p\), and \(L/K\) a primitive extension of degree \(p^n\) for \(n\geq 2\). The author constructs a family of monogenic \(K\)-Hopf algebras \(H\) of dimension \([L:K]=p^n\) and shows that each makes \(L\) into a Hopf Galois object. Namely, \(H=k[t]/(t^{p^n})\) with various comultiplications by using Witt vector addition polynomials \(w_d\) for each positive integer \(d\) as defined in [\textit{N. Jacobson}, Lectures in abstract algebra. Vol. 3: Theory of fields and Galois theory. Reprint of the 1951-1964 Van Nostrand ed. Berlin: Springer-Verlag (1976; Zbl 0322.12001)]. Let \(r\) be a positive integer less than \(n\). For \(\eta\in K^\times\), \(\eta_1=\eta\), \(\eta_i=\eta^{p^{1-i}}\eta_{i-1}^{p^r}\) for \(i\in\mathbb Z^+\), \(d=[n/r]-1\). Define \(\triangle\colon H\to H\otimes_kH\) by \(\triangle(t)=S_d((\eta_dt^{p^{d_r}}\otimes_k1,\ldots,\eta_1t^{p^r}\otimes_k1,t\otimes_k1);(1\otimes_k\eta_dt^{p^{d_r}},\ldots,1\otimes_k\eta_1t^{p^r},1\otimes_kt))\). Then \(H\) denoted by \(H_{n,r,\eta}\) is a \(k\)-Hopf algebra with counit \(\varepsilon(t)=0\) and antipode \(\lambda(t)=-t\). Similarly, for an \(f\in K^\times\), \(f_1=f^{1/p}\), \(f_i=f^{p^{-i}}f_{i-1}^{p^r}\), \(i\geq 2\), a \(K\)-Hopf algebra \(K[t]/(t^{p^n})\) denoted by \(H_{n,r,f}\) is defined.NEWLINENEWLINE Theorem 1. \(H_{n,r,f}\cong H_{n,r,f'}\) for \(f,f'\in K^\times\) if and only if \(f/f'\in(K^\times)^{p^{r+1}-1}\).NEWLINENEWLINE Theorem 2. Let \(L=K(x)\), \(x^{p^n}=b\in K\). If \(H = H_{n,r,f}\) for some choice of a positive integer \(r\) less than \(n\) and \(f\in K^\times\), then \(L\) is an \(H\)-Galois object. Thus \(L/K\) has infinite number of \(K\)-Hopf algebras \(H\) such that \(L\) is an \(H\)-Galois object. Moreover, the structure of \(H\) (\(=H_{n,r,f}^*\)) and the coaction of \(H\) on \(L\) in the case \(r=n-1\) are described such that \(L/K\) is an \(H\)-Galois extension. It is also shown that the above construction of Hopf algebras can be adapted to general modular extensions.
scientific article; zbMATH DE number 6359495