From the reviews: "Gabriel Daniel Villa Salvador's book Topics in the Theory of Algebraic Function Fields was recently translated into English and published by Birkhauser (the original version was published in Mexico in 2003). In the book, Salvador covers a large amount of material at a wide range of levels of difficulty, and as such his book could be used in a wide variety of settings... The entire book is very well written and Villa Salvador includes many examples to both illustrate and motive the theory being presented. There are also a handful of exercises in each chapter... The study of function fields is a beautiful area of mathematics which seems to be ever-increasing in importance, and this book is a wonderfully well-written introduction to the area." -MAA Reviews "The book is intended as a self-contained introduction to the theory of algebraic function fields in one variable. It is based on the original Spanish edition, published in 2003. ... It also includes exercises and an appendix on group cohomology. Most of the topics treated are fairly standard for textbooks on algebraic function fields. ... The book contains numerous exercises." (Manfred Lochter, Mathematical Reviews, Issue, 2007 i) "This book is the algebraic-arithmetic approach studying function fields as part of the algebraic theory of numbers. ... The text is written in a very lucid and comprehensive style suitable for a first-year graduate course on algebraic number theory. ... It is a beautiful book showing how different algebraic theories fit together creating the huge building of algebraic function fields." (H. Mitsch, Monatshefte fur Mathematik, Vol. 153 (4), April, 2008)

• Algebraic and Numerical Antecedents.- Algebraic Function Fields of One Variable.- The Riemann-Roch Theorem.- Examples.- Extensions and Galois Theory.- Congruence Function Fields.- The Riemann Hypothesis.- Constant and Separable Extensions.- The Riemann-Hurwitz Formula.- Cryptography and Function Fields.- to Class Field Theory.- Cyclotomic Function Fields.- Drinfeld Modules.- Automorphisms and Galois Theory.

From the reviews: "Gabriel Daniel Villa Salvador's book Topics in the Theory of Algebraic Function Fields was recently translated into English and published by Birkhauser (the original version was published in Mexico in 2003). In the book, Salvador covers a large amount of material at a wide range of levels of difficulty, and as such his book could be used in a wide variety of settings... The entire book is very well written and Villa Salvador includes many examples to both illustrate and motive the theory being presented. There are also a handful of exercises in each chapter... The study of function fields is a beautiful area of mathematics which seems to be ever-increasing in importance, and this book is a wonderfully well-written introduction to the area." -MAA Reviews "The book is intended as a self-contained introduction to the theory of algebraic function fields in one variable. It is based on the original Spanish edition, published in 2003. ... It also includes exercises and an appendix on group cohomology. Most of the topics treated are fairly standard for textbooks on algebraic function fields. ... The book contains numerous exercises." (Manfred Lochter, Mathematical Reviews, Issue, 2007 i) "This book is the algebraic-arithmetic approach studying function fields as part of the algebraic theory of numbers. ... The text is written in a very lucid and comprehensive style suitable for a first-year graduate course on algebraic number theory. ... It is a beautiful book showing how different algebraic theories fit together creating the huge building of algebraic function fields." (H. Mitsch, Monatshefte fur Mathematik, Vol. 153 (4), April, 2008)

• Algebraic and Numerical Antecedents.- Algebraic Function Fields of One Variable.- The Riemann-Roch Theorem.- Examples.- Extensions and Galois Theory.- Congruence Function Fields.- The Riemann Hypothesis.- Constant and Separable Extensions.- The Riemann-Hurwitz Formula.- Cryptography and Function Fields.- to Class Field Theory.- Cyclotomic Function Fields.- Drinfeld Modules.- Automorphisms and Galois Theory.

From the reviews: "Gabriel Daniel Villa Salvador's book Topics in the Theory of Algebraic Function Fields was recently translated into English and published by Birkhauser (the original version was published in Mexico in 2003). In the book, Salvador covers a large amount of material at a wide range of levels of difficulty, and as such his book could be used in a wide variety of settings... The entire book is very well written and Villa Salvador includes many examples to both illustrate and motive the theory being presented. There are also a handful of exercises in each chapter... The study of function fields is a beautiful area of mathematics which seems to be ever-increasing in importance, and this book is a wonderfully well-written introduction to the area." -MAA Reviews "The book is intended as a self-contained introduction to the theory of algebraic function fields in one variable. It is based on the original Spanish edition, published in 2003. ... It also includes exercises and an appendix on group cohomology. Most of the topics treated are fairly standard for textbooks on algebraic function fields. ... The book contains numerous exercises." (Manfred Lochter, Mathematical Reviews, Issue, 2007 i) "This book is the algebraic-arithmetic approach studying function fields as part of the algebraic theory of numbers. ... The text is written in a very lucid and comprehensive style suitable for a first-year graduate course on algebraic number theory. ... It is a beautiful book showing how different algebraic theories fit together creating the huge building of algebraic function fields." (H. Mitsch, Monatshefte fur Mathematik, Vol. 153 (4), April, 2008)

• Algebraic and Numerical Antecedents.- Algebraic Function Fields of One Variable.- The Riemann-Roch Theorem.- Examples.- Extensions and Galois Theory.- Congruence Function Fields.- The Riemann Hypothesis.- Constant and Separable Extensions.- The Riemann-Hurwitz Formula.- Cryptography and Function Fields.- to Class Field Theory.- Cyclotomic Function Fields.- Drinfeld Modules.- Automorphisms and Galois Theory.