Multinorm one tori are stably rational when the gcd d of factor degrees is 1, with a retract-rationality criterion, failure conditions for d>1, and equivalent stable-rationality conditions when splitting occurs over Galois extensions whose groups have cyclic Sylow subgroups or are dihedral.
Kang, Retract rational fields, J
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The rationality problem for multinorm one tori, II
Multinorm one tori are stably rational when the gcd d of factor degrees is 1, with a retract-rationality criterion, failure conditions for d>1, and equivalent stable-rationality conditions when splitting occurs over Galois extensions whose groups have cyclic Sylow subgroups or are dihedral.