Elementos normales y k-normales sobre cuerpos finitos.

Normal elements were defined, and their existence proved, more than
150 years ago. However, due to their many applications, they have
been vastly studied in the last 30 years. On the other hand, k-normal
elements that generalize normal elements were only introduced a few
years ago.
First we briefly give an account of basic properties and results on
normal elements including existence and number of normal elements.
We briefly focus on how to operate with normal basis, and we discuss
how to find normal elements. It turns out that not all normal elements
behave in the same way, the optimal normal elements being
preferable. These special elements are directly related to Gauss periods
in finite fields. Since optimal normal elements only exist on some
extension fields, the study of low complexity normal elements is
relevant.
Then, we define k-normal elements and survey their main properties.
We comment on their existence and number, as well as on the
existence of elements that are k-normal and primitive at the same time.
We conclude giving some open problems.