Packet 8: Tossup 7

A finite extension of the rationals has a property of this name if and only if it is contained in a cyclotomic field. The Mordell-Weil theorem applies to structures like elliptic curves that have a property of this name when considered as varieties. For a group to be solvable, each quotient group in its subnormal series must have a property (15[1])of this name. If a group has a property of this name, its torsion elements form a subgroup, and it equals its own (*) center. (10[1])By (10[1])a fundamental theorem, finite groups with that property of this name (10[1])are the direct sum of cyclic groups of prime power order. If a group (10[1])has a property of this name, its Cayley table is symmetric (10[2])and any conjugation is the identity; such groups include Z/nZ (“Z-mod-n-Z”). For 10 points, a group whose binary operation is commutative has what property (10[3])named for a Norwegian (10[2])mathematician? ■END■ (10[3])

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