From the reviews of the second edition: "This second and considerably enlarged edition reflects the progress made in field arithmetic during the past two decades. ... The book also contains very useful introductions to the more general theories used later on ... . the book contains many exercises and historical notes, as well as a comprehensive bibliography on the subject. Finally, there is an updated list of open research problems, and a discussion on the impressive progress made on the corresponding list of problems made in the first edition." (Ido Efrat, Mathematical Reviews, Issue 2005 k) "The goal of this new edition is to enrich the book with an extensive account of the progress made in this field ... . the book is a very rich survey of results in Field Arithmetic and could be very helpful for specialists. On the other hand, it also contains a large number of results of independent interest, and therefore it is highly recommendable to many others too." (Roberto Dvornicich, Zentralblatt MATH, Vol. 1055, 2005) From the reviews of the third edition: "The book give an introduction to the arithmetic of fields that is fairly standard, covering infinite Galois theory, profinite groups, extension of valued fields, algebraic function fields ... and an introduction to affine and projective curves providing a geometric interpretation for results formulated in the language of function fields. ... It could be used a text for graduate students entering the field, since the material is so well organized, even including exercises at the end of every chapter." (Felipe Zaldivar, MAA Online, December, 2008)