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Bases and the distribution of irreducible and primitive polynomials over finite fields. (English) Zbl 0877.11068

Gollmann, Dieter (ed.), Applications of finite fields. Based on the proceedings of the IMA conference, London, UK, July 5–7, 1994. Oxford: Clarendon Press. Inst. Math. Appl. Conf. Ser., New Ser. 59, 1-18 (1996).
A wide ranging survey of recent progress on the existence and construction of bases and irreducible and primitive polynomials, under certain conditions, is given. Polynomial, dual and self-dual bases are considered. The concepts of weakly self-dual and almost weakly self-dual bases are discussed with conditions for their existence given. The notion of the excess of the transformation matrix, over the matrix size, \(n\), is noted. The construction of normal bases is discussed and the conjecture that there always exists a primitive normal polynomial of degree \(n \geq 2\) over \(F_q\) with trace \(a\) made. The notion of completely free elements in extension fields is introduced and known construction techniques for them given. It is conjectured that for \(n \geq 2\) there always exists a completely normal primitive polynomial of degree \(n\) over \(F_q\). A variety of interesting questions on the existence and construction of irreducible and primitive polynomials are considered and, in particular, constructions of infinite sequences of irreducible and N-polynomials (independent roots) are given. Many other aspects of polynomials and bases are touched upon, including a brief note on applications of some of the concepts.
For the entire collection see [Zbl 0868.00043].

MSC:

11T06 Polynomials over finite fields
11T30 Structure theory for finite fields and commutative rings (number-theoretic aspects)
11Y16 Number-theoretic algorithms; complexity
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