A new criterion on \(k\)-normal elements over finite fields (Q1997178)

In this paper the authors study \(k\)-normal elements over finite fields. This concept determines a generalization of normal elements. Indeed, for given integers \(k, n\) such that \(0\leq k <n\), a \(k\)-normal element \(\alpha\in \mathbb F_{q^n}^*\) over \(\mathbb F_q\) is an element such that \(\dim_{\mathbb F_q}\langle \alpha, \alpha^q,\ldots,\alpha^{q^{n-1}}\rangle_{\mathbb F_q}=n-k\). In other words, \(0\)-normality coincides with normality. The main contribution of this paper is the introduction of a new criterion for an element in \(\mathbb F_{q^n}^*\) to be \(k\)-normal over \(\mathbb F_q\) when \(\gcd(q,n)=1\), which is based on the idempotent decomposition of \(\mathbb F_{q}[x]/(x^n-1)\). The proof relies on the existence of a system of orthogonal minimal idempotents of \(\mathbb F_q[x]/(x^n-1)\) and on the properties of linearized \(q\)-polynomials.