By Chern S.S., Li P., Cheng S.Y., Tian G. (eds.)

Those chosen papers of S.S. Chern speak about subject matters equivalent to essential geometry in Klein areas, a theorem on orientable surfaces in 4-dimensional house, and transgression in linked bundles

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**Example text**

8 The sets Br (K |k) and Br (k) equipped with the above product operation are abelian groups. 32 Central simple algebras and Galois descent Before proving the proposition, we recall a notion from ring theory: the opposite algebra A◦ of a k-algebra A is the k-algebra with the same underlying k-vector space as A, but in which the product of two elements x, y is given by the element yx with respect to the product in A. If A is central simple over k, then so is A◦ . Proof Basic properties of the tensor product imply that the product operation is commutative and associative.

In (2), one chooses x = α and y as in the proposition. 6 Reduced norms and traces 37 √ an arbitrary degree m cyclic Galois extension K |k in the form K = k( m a), as in the corollary above. 13 (1)) shows that a cyclic Galois extension of degree p in characteristic p > 0 is generated by a root of some polynomial x p − x − a. In the previous chapter we have seen that the class of a nonsplit quaternion algebra has order 2 in the Brauer group. More generally, the class of a cyclic division algebra (a, b)ω as above has order m; we leave the veriﬁcation of this fact as an exercise to the reader.

Now observe that for every ﬁnite ﬁeld extension K of k contained in k, ¯ ¯ the inclusion K ⊂ k induces an injective map A ⊗k K → A ⊗k k and A ⊗k k¯ arises as the union of the A ⊗k K in this way. Hence for a sufﬁciently large ﬁnite extension K |k contained in k¯ the algebra A ⊗k K contains the elements ¯ via e1 , . . , en 2 ∈ A ⊗k k¯ corresponding to the standard basis elements of Mn (k) ∼ ¯ ¯ the isomorphism A ⊗k k = Mn (k), and moreover the elements ai j occurring in the relations ei e j = ai j ei deﬁning the product operation are also contained in K .