By Wolfgang Franz

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Example text

We see that the maps x > ax (for a j& 0) Suppose b by also either preserve or reverse the ordering. 16. // a field k is weakly ordered and has more than two elements, then it is an ordered field. Hilbert has constructed an ordered non-commutative field and we shall present his example. F be any field and a an automorphism of F. We denote by 9 a the image of an element a t F under <7 and by a* its image under the automorphism or*. We construct an extension field k of F consisting of all formal Let power series which contain only a finite number of non-zero coi; in other words we assume that for each can be found such that a, = for i < N.

Let ofG. G meet it again in the theory of Banach spaces of analysis. 8. The set of all x t G for which the set xS is the same as the set Sx is called the normalizer N s of the set S. The set which xs = sx for all s t S is called the centralizer Z s of S. The set of all x e G which commute with all elements of G, in other words the centralizer Z of G is called the center of G. of all x c G for 30 Ns GEOMETRIC ALGEBRA and Z R are subgroups G and of invariant (normal) subgroup of the center is a commutative (?.

The number of distinct images

, or . l . , : . , N s ) of the normaliser of S. equal to the index (G Of special importance is the case where S consists of one element = {a}. Then each

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Algebraic topology by Wolfgang Franz


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