Special invariant subspaces of a vector space over \(Z/lZ\). (English) Zbl 0737.11027

Skula, Ladislav

Arch. Math., Brno 25, No. 1-2, 35-46 (1989).

Let \(\ell\) be an odd prime, \(N=(\ell-1)/2\), and \(V=(\mathbb{Z}/\ell\mathbb{Z})^ N\). The author constructs an operator \(S\) on \(V\) and shows that each quadratic nonresidue \(\hbox{mod} \ell\) occurs as an eigenvalue of \(S\). Let \(G\) be a cyclic group of order \(\ell-1\) generated by an element \(s\) and let \({\mathfrak R}(\ell)\) be the group ring \(\mathbb{Z}_ \ell[G]\). Let \({\mathfrak R}^ -(\ell)\) be the elements in the group ring annihilated by multiplication by \(1+s^ N\). The author establishes a bijective correspondence between the ideals of \({\mathfrak R}^ -(\ell)\) and the \(S\)- invariant subspaces of \(V\). For \(\ell<1000\) he states that the coordinate matrix for the subspace corresponding to the Stickelberger ideal has a special normal form.

Reviewer: L.Washington (College Park)

MSC:

11R18

Cyclotomic extensions

Keywords:

cyclotomic fields; cyclic group; group ring; invariant subspaces; Stickelberger ideal