Some Non-commutative Methods in Algebraic Number Theory

Some time between the years 1930–32 I heard Emmy cry out: " 1 – S = 2 if S = - 1." What she meant was, of course, that the symbolic power 1 – S implies squaring if S is the automorphism given by the inverse. Many times I heard her say, in many contexts: "Das muss hyperkomplex bewiesen werden," using the word hyperkomplex as an adverb. Both of these utterances were crucial for the work of Emmy that fits into the title of this article. Their implications illuminate a vast area of methods, formulations, new ideas.