Higher local and global class field theory was developed by A. Parshin, Kazuya Kato, Ivan Fesenko, Spencer Bloch, Shuji Saito and other mathematicians.
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Higher dimensional class field theory was pioneered by A.N. Parshin in positive characteristic and K. Kato, I. Fesenko, Sh.
Shafarevich–Weil theorem about the fundamental class in class field theory