Preface
Much of the second and third year undergraduate course in mathematics (as well as
some graduate work) was covered by Volumes 2 and 3 of my book on algebra, now
out of print. 1 So I was very pleased when Springer Verlag offered to bring out a new
version of these volumes. The present book is based on both these volumes, comple-
mented by the definitions and basic facts on groups and rings. Thus the volume is
addressed to students who have some knowledge of linear algebra and who have
met groups and fields, though all the essential facts are recalled here. My overall
aim has been to present as many of the important results in algebra as would
conveniently fit into one volume. It is my hope to collect the remaining parts of
Volumes 2 and 3 into a second book, more oriented towards applications. 2
Apart from chapters on groups (Chapter 2), rings and modules (Chapters 4, 5 and 6)
and fields (Chapters 7 and 11), a number of concepts are treated that are less central
but nevertheless have many uses. Chapter 1, on set theory, deals with countable and
well-ordered sets, as well as Zorn's lemma and a brief section on graphs. Chapter 3
introduces lattices and categories, both concepts that form an important part of
the language of modern algebra. The general theory of quadratic forms has many
links with ordered fields, which are developed in Chapter 8. Chapters 9 and 10 are
devoted to valuation theory and commutative rings, a subject that has gained in
importance through its use in algebraic geometry.
On a first encounter some readers may find the style of this book somewhat
concise, but they should bear in mind that mathematical texts are best read with
paper and pencil, to work out the full consequences of what is being said and to
check examples. The matter has been well put by Einstein, who said: "Everything
should be explained as far as possible but no further." There are numerous exercises
throughout, with occasional hints (but no solutions), and some historical remarks.
My thanks are due to the staff of Springer Verlag for the efficient way they have
produced this volume.
University College London P.M. Cohn
June 2002
I Algebra, Vol. 2 (2nd edn, 1989) and Vol. 3 (2nd edn, 1991), Wiley and Sons.
2 Further Algebra and Applications, Springer Verlag, London (2003). Referred to in the text as FA.
ix
Much of the second and third year undergraduate course in mathematics (as well as
some graduate work) was covered by Volumes 2 and 3 of my book on algebra, now
out of print. 1 So I was very pleased when Springer Verlag offered to bring out a new
version of these volumes. The present book is based on both these volumes, comple-
mented by the definitions and basic facts on groups and rings. Thus the volume is
addressed to students who have some knowledge of linear algebra and who have
met groups and fields, though all the essential facts are recalled here. My overall
aim has been to present as many of the important results in algebra as would
conveniently fit into one volume. It is my hope to collect the remaining parts of
Volumes 2 and 3 into a second book, more oriented towards applications. 2
Apart from chapters on groups (Chapter 2), rings and modules (Chapters 4, 5 and 6)
and fields (Chapters 7 and 11), a number of concepts are treated that are less central
but nevertheless have many uses. Chapter 1, on set theory, deals with countable and
well-ordered sets, as well as Zorn's lemma and a brief section on graphs. Chapter 3
introduces lattices and categories, both concepts that form an important part of
the language of modern algebra. The general theory of quadratic forms has many
links with ordered fields, which are developed in Chapter 8. Chapters 9 and 10 are
devoted to valuation theory and commutative rings, a subject that has gained in
importance through its use in algebraic geometry.
On a first encounter some readers may find the style of this book somewhat
concise, but they should bear in mind that mathematical texts are best read with
paper and pencil, to work out the full consequences of what is being said and to
check examples. The matter has been well put by Einstein, who said: "Everything
should be explained as far as possible but no further." There are numerous exercises
throughout, with occasional hints (but no solutions), and some historical remarks.
My thanks are due to the staff of Springer Verlag for the efficient way they have
produced this volume.
University College London P.M. Cohn
June 2002
I Algebra, Vol. 2 (2nd edn, 1989) and Vol. 3 (2nd edn, 1991), Wiley and Sons.
2 Further Algebra and Applications, Springer Verlag, London (2003). Referred to in the text as FA.
ix
