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This is an English translation of the monograph "Die Klassenk\"orper derkomplexen Multiplikation" by Max Deuring, published in 1958 as Enzyklop\"adieder Mathematischen Wissenschaften, Band I-2, Heft 10, Teil II by B.G. TeuberVerlagsgesellschaft, Stuttgart. It gives a systematic exposition of theanalytic method in the theory of complex multiplication -- the interrelationsbetween elliptic functions, modular functions and algebraic numbers.  This translation was done as an exercise for learning the German language.The mathematical notation and terminology are kept as closely as possible tothose given in the German original work. No attempt has been made to modernizethe exposition, other than rectifying a (very) few obviously-correctabletypographical errors.

Class Fields of Complex Multiplication: translation of "Die Klassenk\"orper der komplexen Multiplikation" by Max Deuring