# Vortrag

Normal bases in finite fields constitute a vast topic of large theoretical and practical interest. Recently, $k$-normal elements were introduced as a natural extension of normal elements. The questions of the existence and cardinalities of $k$-normal elements comprise an active research avenue, and in full generality remain open problems. In this talk I will first give a description of normal elements, some key results on them, and their significance. I will then define $k$-normal elements and provide some results on their existence and numbers, along with brief outlines of the methods employed in the proofs. In particular, a general lower bound for the number of k-normal elements, assuming that they exist, will be formulated. Further, a new existence condition for $k$-normal elements using the general factorization of the polynomial $x^m-1$ into cyclotomic polynomials, will be derived. Finally, an existence condition will be stated for normal elements in a finite field with a non-maximal but high multiplicative order in the group of units. This final result is seen to be closely related to the well-known Primitive Normal Basis Theorem, and is also proven using the same techniques.