Algebra General

What do you think are the best pathways when learning algebra?

When I took a course we basically went straight for Galois theory ASAP (which I heard is unusual). There are so many different ways you can take algebra, so I want to hear what Veeky Forums did or what you'd recommend.

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My undergrad went
1 Intro linear algebra
2 Rings and fields (baby Hungerford), more linear algebra (Axler)
3 Group theory (Armstrong), reflection groups (Humphreys), Galois theory
4 Algebra (Dummit/Foote), Commutative/homological (Eisenbud/Matsumura/etc.), Representation theory (Vinberg)

I think I turned out all right, I think groups could have come earlier though

Are these separate courses?

Yes, with the textbooks used for each one

I probably took an above average number of algebra classes

the best path is Henri lombardi book

Commutative Algebra, Constructive Methods (Finitely generated projective modules).

hlombardi.free.fr/publis/LivresBrochures.html

This voluminous book offers a very special and detailed introduction to
various basic concepts, methods, principles, and results of commutative
algebra. In contrast to most of the numerous other primers in the field, the
authors consequently pursue the constructive viewpoint in commutative
algebra, that is, they endeavour to extricate the explicit algorithmic
approaches to the many existence theorems in this central area of
contemporary mathematical research, thereby revisiting several classical
abstract topics in a highly clarifying, remarkably simplifying, and
methodologically novel way. Very much in the spirit of the great developers
of constructive algebra in the 19th century, first and foremost C. F. Gauss
and L. Kronecker, the authors reveal the algorithmic aspects of such
naturally abstract topics as Galois theory, Dedekind rings, Pr\"ufer rings,
finitely generated projective modules, dimension theory of commutative
rings, and others in the current treatise, and that in a very enlightening,
inspiring and truly unique style of expository presentation. In particular,
a special characteristic of the text is the strong emphasis on the study of
finitely generated projective modules in commutative algebra, which actually
play a significant role as an algebraic version of the notion of vector
bundle in different areas of modern geometry.\par Assuming a basic knowledge
of linear algebra, group theory, elementary number theory as well as the
fundamentals of ring and module theory, the book under review is mainly
geared toward graduate students on the levels M1 and M2 at European
universities, but also researchers, instructors, and theoretical computer
scientists can profit a great deal from the novel ideas and aspects exposed
in this rich source of modern constructive algebra.

\par As for the precise
contents, the book consists of seventeen chapters and an appendix on
constructive logic. Each chapter is divided into several thematic sections,
with a set of related exercises and some bibliographic comments at the
end.\par After a very carefully composed preface, including a rather
detailed description of the contents of the single chapters of the book,
Chapter 1 gives some motivating geometric examples, basically the concepts
of vector bundles and modules of differential forms. Chapter 2 explains the
``local-global principle'' in commutative algebra, together with its
relations to systems of linear equations over commutative rings. Chapter 3
develops the so-called ``method of indeterminate coefficients'' initiated by
Gauss, based on which several existence theorems are then constructively
treated. Modules of finite presentation are introduced in Chapter 4, where
also special rings, Fitting ideals, and resultant ideals are discussed.

Chapter 5 gives a first approach to finitely generated projective modules,
while Chapter 6 deals with strictly finite algebras and Galois algebras,
thereby providing some constructive basic Galois theory as well as
fundamental results on \'etale algebras and separable algebras. Chapter 7
presents another constructive method in (polynomial) algebra, namely the
so-called ``dynamic method''. By this method, the abstract proofs of various
existence theorems in algebra can be made explicit, ranging from Hilbert's
Nullstellensatz to Galois theory over discrete fields. Chapter 8 turns to
flat modules, with particular emphasis on flat ideals, flat algebras, and
faithfully flat algebras. Local rings and their relatives are the main topic
of Chapter 9, where the constructive aspects of decomposable rings as well
as concrete examples of local rings in algebraic geometry are discussed
along the way.\par Chapter 10 gives a more in-depth treatment of finitely
generated modules, together with their relations to algebro-geometric
constructions such as Grassmannians, Grothendieck groups, Picard groups, and
others. Chapter 11 depicts the appearance of distributive lattices and
reticular groups in the context of constructive commutative algebra, and the
subsequent Chapter 12 provides a very explicit and versatile description of
the structure theory of Pr\"ufer rings and of Dedekind rings. The notion of
Krull dimension of commutative rings is dealt with in Chapter 13. This
includes a constructive definition of that concept just as a description of
integral ring extensions, dimension theory of morphisms, valuative dimension
theory, and a Krull dimension theory of distributive lattices. Chapter 14
explains the fundamental results concerning the classical problem of
determining the number of generators of a finitely generated module. The

authors give constructive, fairly elementary versions of the relevant
theorems due to L. Kronecker, H.Bass, J.-P.Serre, O. Forster, and R. Swan,
respectively, along with a discussion of the so-called Heitmann dimension of
a commutative ring. Chapter 15 presents a number of important constructive
methods which are directly related to the local-global principle in
commutative algebra, and Chapter 16 turns to the study of finitely generated
projective modules over a polynomial ring. The authors present here
constructive approaches to the fundamental results by Traverso-Swan,
Quillen-Vaserstein, Horrocks, Suslin, Coquand, Lombardi, Quitt\'e, and
others in a very detailed manner, thereby using their own research
contributions effectively. Chapter 17 is devoted to an entirely constructive
proof of A. Suslin's famous stability theorem in the case of a discrete
ground field, where the concrete local-global principle of {\it R. Rao}
(1985) appears as a crucial ingredient.\par Apart from an instructive
appendix on some principles of constructive logic, a rich bibliography of
187 references, a table of theorems proved in the book, and two, very
carefully compiled indices of both notions and terms provide an utmost
useful service to the reader. The incredible wealth of more than 350
exercises and problems represent another outstanding feature of this
marvelous, weighty and rather unique textbook in commutative algebra. In
fact, these well-arranged exercises, together with their helpful hints for
solution, form an essential component of the book, which provides an
abundance of additional important material and examples at the same
time.\par All together, the book under review must be seen as an invaluable
replenishment of the existing textbook literature in commutative algebra.

My undergrad looks like this (partly self-taught, hence the weird order):
Group theory, lin alg, rings and fields, galois theory, rep theory. In that order.
Books I used were some notes, Hoffman-Kunze [sic?], Dummit-Foote (for the modules part of rings and fields), some terrible book not worth remembering for Galois theory, and Etingof and the first ~3 chaps of Weibel.