By Kanat Abdukhalikov (auth.), Dieter Jungnickel, Harald Niederreiter (eds.)

The 5th foreign convention on Finite Fields and purposes **F**q5 held on the college of Augsburg, Germany, from August 2-6, 1999 persevered a sequence of biennial overseas meetings on finite fields. The complaints rfile the gradually expanding curiosity during this subject. Finite fields have an inherently attention-grabbing constitution and are very important instruments in discrete arithmetic. Their purposes variety from combinatorial layout idea, finite geometries, and algebraic geometry to coding concept, cryptology, and medical computing. a very fruitful point is the interaction among concept and purposes which has ended in many new views in study on finite fields. This interaction has consistently been a dominant topic in **F**q meetings and used to be a great deal in proof at **F**q5. The lawsuits replicate this, and provide an up to date selection of surveys and unique study articles through top specialists within the region.

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**Example text**

Coulter, On the evaluation of a class of Weil sums in characteristic 2, New Zealand J. Math. 28 (1999), 171-184. 4. P. G. Ostrom, Planes of order n with collineation groups of order n 2 , Math. Z. 103 (1968), 239-258. 5. L. , Amer. Math. Monthly 95 (1988), 243-246. 6. R Lidl and H. Niederreiter, Finite Fields, Encyclopedia Math. , vol. 20, Addison-Wesley, Reading, 1983, (now distributed by Cambridge University Press). 42 Blokhuis, Coulter, Henderson and O'Keefe 7. J. Patarin, Hidden fields equations (HFE) and isomorphisms of polynomials (IP): two new families of asymmetric algorithms, Advances in Cryptology - Eurocrypt '96 (D.

Further results, obtained for fields of characteristic i= 2, are often formulated in terms of cohomological invariants. This is the subject matter of §6, where the notion of G -discriminant is introduced (this invariant is implicit in [5]). For the fields of cohomological dimension 1 (for instance, finite fields) the G-discriminant is a complete invariant. In the case of extensions of finite fields, the G-discriminant can take at most two values, hence there are at most two possibilities for the isomorphism class of the G-trace form.

X + .... e. satisfying (1). 2, p. 4, p. 31]. Our result shows that all the formal groups that are rational functions are equivalent to either x + y or x + y + xy with the equivalence defined by rational functions, instead of formal power series. Note that determining all the formal groups that are polynomials is much easier as indicated by the exercise in [4, p. 1 in [3, p. 1 in the next section. Our interest in this paper was motivated by the work in [1,2,5] where it is desirable to construct irreducible polynomials of higher degrees from those of lower degrees.