Specialization of Quadratic and Symmetric Bilinear Forms by Manfred Knebusch

Author:Manfred Knebusch
Language: eng
Format: epub
Publisher: Springer London, London

(2)If φ is of inner type and , then is also regular. On the other hand, if φ is of outer type, then is regular over the composite .

Example. The projective standard tower of φ is regular by Theorem 2.53. The affine standard tower of φ is likewise regular.

Now let be a regular generic splitting tower of φ and let be a φ-conservative field extension of k. Using and L we want to construct a generic splitting tower of .

Definition 2.57.

For every we construct a field composite of and L over k as follows: if , or if and φ is of inner type, then is free of zero divisors and we let be the quotient field of this ring, i.e. the uniquely determined free composite of K and L over k. If and φ is of outer type, we distinguish the cases where is of outer/inner type.

First, let be of outer type. Then is a field. Let be the free composite of with over , i.e. again the quotient field of . Finally, let be of inner type. Now can be embedded over k in L in two ways. We choose one such embedding and set .

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