By Philippe Gille

This e-book is the 1st finished, smooth creation to the speculation of important basic algebras over arbitrary fields. ranging from the fundamentals, it reaches such complex effects because the Merkurjev-Suslin theorem. This theorem is either the fruits of labor initiated via Brauer, Noether, Hasse and Albert and the place to begin of present learn in motivic cohomology conception through Voevodsky, Suslin, Rost and others. Assuming just a reliable history in algebra, yet no homological algebra, the e-book covers the elemental thought of critical easy algebras, tools of Galois descent and Galois cohomology, Severi-Brauer kinds, residue maps and, eventually, Milnor K-theory and K-cohomology. The final bankruptcy rounds off the idea by means of featuring the consequences in confident attribute, together with the theory of Bloch-Gabber-Kato. The booklet is acceptable as a textbook for graduate scholars and as a reference for researchers operating in algebra, algebraic geometry or K-theory.

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**Example text**

Yr are D-linearly independent (because so are 1 ⊗ w1 , . . , 1 ⊗ wr ), so they form a D-basis of J . As J is a two-sided ideal, for all d ∈ D we must have d −1 yi d ∈ J for 1 ≤ i ≤ r , so there exist βil ∈ D with d −1 yi d = βil yl . e. αi j ∈ k as D is central. This means that J can be generated by elements of K (viewed as a k-subalgebra of D ⊗k K via the embedding w → 1 ⊗ w). As K is a ﬁeld, we must have J ∩ K = K , so J = D ⊗k K . This shows that D ⊗k K is simple. 2. 7 imply that A ⊗k k¯ ∼ = Mn (k) ¯ n.

En be the standard basis of K n . Mapping a matrix M ∈ I1 to Me1 induces an isomorphism I1 ∼ = K n of n K -vector spaces, and thus λ induces an automorphism of K . As such, it is given by an invertible matrix C. We get that for all M ∈ Mn (K ), the endomorphism of K n deﬁned in the standard basis by λ(M) has matrix C MC −1 , whence the lemma. 2 The automorphism group of Mn (K ) is the projective general linear group PGLn (K ). Proof There is a natural homomorphism GLn (K ) → Aut(Mn (K )) mapping C ∈ GLn (K ) to the automorphism M → C MC −1 .

1 we see that there is an irreducible polynomial f ∈ k[x] and a k-algebra homomorphism k[x]/( f ) → D whose image contains d. But k being algebraically closed, we have k[x]/( f ) ∼ = k. 2 Splitting ﬁelds The last corollary enables one to give an alternative characterization of central simple algebras. 1 Let k be a ﬁeld and A a ﬁnite dimensional k-algebra. Then A is a central simple algebra if and only if there exist an integer n > 0 and a ﬁnite ﬁeld extension K |k so that A ⊗k K is isomorphic to the matrix ring Mn (K ).