Submission: 2007, Apr 15
For any $n\ge 3$ we give numerous examples of central division algebras of exponent $2$ and index $2^n$ over fields, which do not decompose into a tensor product of two nontrivial central division algebras, and which are sums of $n+1$ quaternion algebras in the Brauer group of the field. Also for any $n\ge 3$ and any field $k_0$ we construct an extension $F/k_0$ and a multiquadratic extension $L/F$ of degree $2^n$ such that for any proper subextensions $L_1/F$ and $L_2/F$ $$W(L/F)\not= W(L_1/F)+W(L_2/F),\ \ \ _2 Br (L/F)\not=_2 Br (L_1/F)+ _2 Br (L_2/F).$$
2000 Mathematics Subject Classification: 16K20, 14H05
Keywords and Phrases: Brauer group, division algebra, conic, Laurent series field
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