Reference for book on fundamental abstract algebra topics

Question

Can anybody suggest a good book on the topics listed below? A single book would be preferable. Thanks.

Groups, subgroups, normal subgroups,cosets,Lagrange’s theorem, rings and their properties, commutative rings, integral domains and fields, subrings, ideals and their elementary properties. Vector spaces, subspaces and their properties,linear independence and dependence of vectors, matrices, rank of a matrix, reduction to normal forms, linear homogeneous and non-homogenous equations, Cayley-Hamilton theorem, characteristic roots and vectors. DeMoivre’s theorem, relation between roots and coefficient of $n$-th degree equation, solution to cubic and biquadratic equations, transformation of equations.

Answer 1

I can't confirm that every single topic above is included, but Serge Lang's Algebra is an extremely comprehensive tome for all algebra topics up to and including first year graduate-level algebra. It's not exactly readable, but it has a ton of exercises and gives you all the logical steps you need to explore these topics.

Answer 2

These subjects are not all contained in what most people would think of as abstract algebra... I don't think it's really possible to fulfil your "one book only" request.

Groups and rings are definitely abstract algebra. Any introductory abstract algebra book will cover them. A Book Of Abstract Algebra is quite good for a first pass, but not as comprehensive as it could be.

Vector spaces and matrices are Linear Algebra. I very highly recommend Jim Hefferson's book, which is available as a free PDF on his website as well as being very cheap on paper, for a textbook. I can testify that it covers everything on your list about vector spaces and matrices, and so will any introductory Linear Algebra book, if Hefferson's doesn't suit you.

The stuff about equations and roots is very much classical algebra. I am not aware of any book that focuses specifically on those topics because they're not very fashionable these days. However, I have found that the introductory chapters of Galois Theory textbooks often cover exactly what you're looking for (and in an elementary way), so you might have some luck there. If nothing else, what you've listed here is elementary enough that you should be able to sort yourself out with online resources.