By Tammo tom Dieck

This ebook is a jewel– it explains very important, priceless and deep issues in Algebraic Topology that you just won`t locate in other places, rigorously and in detail."""" Prof. Günter M. Ziegler, TU Berlin

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Let N be the splitting field over F of {min( F, exi) : 1 ~ i ~ n}. 9 since each min(F, exi) is separable over F. Moreover, K ~ N. By the fundamental theorem, the intermcdiate fields of N / F are in 1-1 correspondence with the subgroups of the finite group Gal(N/ F). Any finite group has only finitely many subgroups, so N j F has only finitely many intermediate fields. In particular, K / F has only finitely many intermediate fields. Therefore, K = F(ex) for some 0: by the primitive element theorem.

O:T be the distinct roots of min( F, 0:). Then, since 0: is separable over F, we have [L : F] = r. By the isomorphism extension theorem, there are Ti E Gal(K/ F) with Ti(O:) = O:i. The cosets TiH are then distinct, since if Ti-1T,j E H = Gal(K/ L), then (Ti- 1 Tj)(0:) = 0:; hence, O:i = Ti(O:) = Tj(O:) = O:j. Let G = Gal(K/ F). l We now extend the concept of separability to field elements and field extensions. 7 Let K be an extension field of F and let 0: E K. Then is separable over F if min( F, 0:) is separable over F.

Beyond this, we can give a proof using group theory and Galois theory. We point out the group theoretic fact we need: If G is a finite group whose order is a power of a prime p, then any maximal subgroup of G has index p in G. 4 of Appendix C. 13 Let f(x) E JR[x]. 1. If f(x) = x 2 - a for some a > 0, then f has a TOot in JR. Therefore, every nonnegative real number has a real square root. 2. If deg(f) is odd, then f has a TOot in JR. Conseq'Ltently, the only odd degree extension of JR is JR itself.

### Algebraic Topology and Tranformation Groups by Tammo tom Dieck

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