Relative normal bases of positive characteristic
This paper deals with integral normal bases (INB) of tame ``cyclotomic'' extensions \(K\) of the rational function field \(Q=\mathbb{F}_q(T)\). More precisely, the extension \(K\) is obtained by adjoining the \(P\)-division points of the Carlitz module to \(Q\), for any irreducible polynomial \(P\in\mathbb{F}_q[T]\), and the issue is whether an intermediate layer \(N/k\), \(Q\subset k \subset N \subset k\), admits an INB, that is, whether the projective \(O_k[\Delta]\)-module \(O_N\) is free. (Here \(\Delta\) is the Galois group of \(N/k\).) The extreme case \(K/Q\) is well understood, just like its counterpart in characteristic zero: INB's do exist and are afforded by Thakur's analogue of Gauss sums. Lowering the top field from \(K\) to \(N\) preserves the existence of an INB, but the effect of raising the bottom field from \(Q\) to \(k\) is surprisingly difficult to understand. In the number field setting \(Q\) is replaced by the rationals \(\mathbb{Q}\) and \(K\) by \(\mathbb{Q}(\zeta_p)\), where \(p\) is an odd prime. In this context, Brinkhuis showed that the only choice of \(k\) such that \([K:k]\) is prime and \(K/k\) has an INB is \(k=K^+\). The author of the present paper proves results for the function field case which point in the same direction. In Theorems 3.5 and 4.6 he gives several sufficient conditions for \(N/k\) to be without INB. As an application, he obtains the following interesting consequences: If the characteristic of \(Q\) is not 2 and \(k\) is the quadratic subfield of \(K/Q\), then \(K/k\) has INB if and only if the polynomial \(P\in\mathbb{F}_q[T]\) defining \(K/Q\) is of degree at most 2. The proof of Theorem 3.5 uses the idea of Brinkhuis that the injectivity of a certain map provides a cohomological obstruction for the existence of integral normal bases, while the proof of 4.6 is more explicit, actually showing the nonprincipality of certain resolvent ideals. In the number field case discussed above, the reviewer, \textit{D. Replogle}, \textit{K. Rubin}, and \textit{A. Srivastav} [J. Number Theory 79, 164-173 (1999; Zbl 0941.11044)] raised and partially answered the question whether \(K/k\) has a weak integral normal basis, that is, \(MO_K\) is \(M\)-free over the maximal order \(M\) in \(k[\text{Gal}(K/k)]\). In the function field case considered above, the order of Gal\((K/Q)\) is already a unit in \(Q\), so \(M\) equals the group ring \(O_k[\text{Gal}(K/k)]\), and hence weak integral normal bases are just the same as integral normal bases, so the analog of the reviewer's question is void in this situation.
scientific article; zbMATH DE number 1943324
scientific article