Multiplicative Invariant Fields of Dimension ≤6

The finite subgroups of G L 4 ( Z ) GL_4(\mathbb {Z}) are classified up to conjugation in Brown, Büllow, Neubüser, Wondratscheck, and Zassenhaus (1978); in particular, there exist 710 710 non-conjugate finite groups in G L 4 ( Z ) GL_4(\mathbb {Z}) . Each finite group G G of G L 4 ( Z ) GL_4(\mathbb {Z}) acts naturally on Z ⊕ 4 \mathbb {Z}^{\oplus 4} ; thus we get a faithful G G -lattice M M with r a n k Z M = 4 \mathrm {rank}_\mathbb {Z} M=4 . In this way, there are exactly 710 710 such lattices. Given a G G -lattice M M with r a n k Z M = 4 \mathrm {rank}_\mathbb {Z} M=4 , the group G G acts on the rational function field C ( M ) ≔ C ( x 1 , x 2 , x 3 , x 4 ) \mathbb {C}(M)≔\mathbb {C}(x_1,x_2,x_3,x_4) by multiplicative actions, i.e. purely monomial automorphisms over C \mathbb {C} . We are concerned with the rationality problem of the fixed field C ( M ) G \mathbb {C}(M)^G . A tool of our investigation is the unramified Brauer group of the field C