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

The 5th overseas convention on Finite Fields and functions **F**q5 held on the college of Augsburg, Germany, from August 2-6, 1999 endured a sequence of biennial foreign meetings on finite fields. The complaints record the gradually expanding curiosity during this subject. Finite fields have an inherently interesting constitution and are vital instruments in discrete arithmetic. Their purposes variety from combinatorial layout conception, finite geometries, and algebraic geometry to coding conception, cryptology, and clinical computing. a very fruitful element is the interaction among conception and purposes which has resulted in many new views in learn on finite fields. This interaction has continuously been a dominant subject in **F**q meetings and was once a great deal in facts at **F**q5. The complaints mirror this, and provide an up to date choice of surveys and unique examine articles through best specialists within the region.

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The 5th overseas convention on Finite Fields and functions Fq5 held on the college of Augsburg, Germany, from August 2-6, 1999 persisted a sequence of biennial foreign meetings on finite fields. The complaints record the gradually expanding curiosity during this subject. Finite fields have an inherently interesting constitution and are vital instruments in discrete arithmetic.

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**Example text**

Let Land L' be two G--Galois algebras over k. If qL qu then dc(L) = do(L'). 4. Let L be a G-Galois algebra over k. If L has a self-dual normal basis over k then do (L) = 1. Note that H I (k,G/G 2) = Hom(n,G/G 2), the set of continous homomorphisms r k -t G / G 2 . As G / G 2 is an elementary abelian 2-group of rank r, this is also the product of r copies of the square classes k* / k*2 . 32 7 Eva Bayer-Fluckiger Groups of Even Order As in section 5, we suppose that k is a field of characteristic i- 2, and G is a finite group.

Further, it is well-known that the polynomial x + y-1 also defines a group on 18' and that the polynomial x + y + xy defines a group on 18' \ { -1 }. It is natural to ask for other such polynomials f (x, y) E 18'[x, y] or even rational functions R( x, y) E 18'( x, y) which define group operations on subsets of F. More generally, one may consider formal power series in two variables and this leads to formal groups of dimension one which are closely related to elliptic curves [6]; see also [4] for formal groups of higher dimension.

Proof. Note T (f(x)) = T (xT(x) = T(x)T(x) + ax 2) + aT(x)2 = (1 + a)T(x)2 for all x E IFqn. Suppose there exists elements x,y E IFqn such that f(x) = f(y). Then x (T(x) + ax) = y (T(y) + ay). Applying the trace function T gives (1 + a) (T(X)2 + T(y)2) = 0, so T(x) = T(y) = t, say. Thus tx + ax 2 = ty + ay2 from which t = a(x implying x + y E IFq. Therefore T(x + y) = n(x + y) = 0. So x = y. 2 cannot have a linearised decompositional factor since their degrees are never divisible by the characteristic.