  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 existence and the number of
$k$-normal elements in a fixed extension of a finite field are both open
problems in full generality, and comprise a promising research avenue. In this
paper, we first formulate a general lower bound for the number of $k$-normal
elements, assuming that they exist. We further derive a new existence condition
for $k$-normal elements using the general factorization of the polynomial
$x^m-1$ into cyclotomic polynomials. Finally, we provide an existence condition
for normal elements in $\fqm$ with a non-maximal but high multiplicative order
in the group of units of the finite field.
