Artin 39;s Algebra

[Barbro Faries]

Emil Artin wrote a book on Geometric Algebra. Is this the same as the geometric algebra of David Hestenes?Please correct me if I'm wrong on this, I think David Hestenes geometric algebra is just clifford algebra. Did David change anything from Clifford' original geometric algebra or is he just trying to make it more well known; it seems to be very useful. Anyway, I am just curious if the textbook Geometric Algebra by Artin is also just clifford algebra or geometric algebra? I don't really know the difference between the two. Thank you.

Artin's book is a much more general course on the connection between algebra and geometry via groups. It does have a section on Clifford algebras, of which "Hestenes geometric algebras" are a special case.

I think Hestenes' principal effect has been to raise the profile of using Clifford algebras practically. I don't think I've ever seen anything about truly new results coming out of the program, but indeed since he raised the profile I think people have been taking a closer look at the Hestenes school perspective.

On the other end of the spectrum, I have seen people say that geometric algebra and the associated geometric calculus is just "for people uncomfortable with basic manifold theory." The whole situation reminds me a little bit of the quaternion/vector-algebra war.

Just remember not to get caught up in hype, because there is a bit of that. Personally, I someday plan to sit down and weigh these things on their virtues, but for now I'm still brushing up my manifold theory :)

Michael Artin's "Algebra" is a textbook that covers various topics in abstract algebra, including group theory. It is commonly used as a reference for undergraduate and graduate level courses in mathematics.

Artin's "Algebra" provides a comprehensive and rigorous treatment of group theory, making it an excellent resource for learning the fundamentals of this branch of mathematics. It includes numerous examples, exercises, and applications to help students understand the concepts and develop problem-solving skills.

While "Algebra" is primarily used as a textbook for courses, it can also be used for self-study. However, it is recommended that the reader have a strong background in mathematics, particularly in linear algebra and abstract algebra, before attempting to study group theory from this book.

Artin's "Algebra" covers a wide range of topics in group theory, including subgroups, homomorphisms, normal subgroups, and factor groups. However, it may not be suitable for advanced topics such as representation theory or Galois theory. It is best to consult with a more specialized textbook for these topics.

Yes, there are various online resources and lecture notes available that can supplement the material covered in "Algebra" for group theory. Additionally, there are other textbooks that may provide a different perspective on the subject and can be used in conjunction with "Algebra" for a more comprehensive understanding of group theory.

Artin published [1] on the related Wedderburn theorem, but Zorn does not cite a publication on the theorem he attributes to Artin in his 1930 paper [2]. Moufang [3] also cites Zorn in her 1935 publication on Artin's theorem as the only published source.

As discussed here, Zorn was Artin's Ph.D. student in Hamburg, Zorn's 1930 Ph.D thesis was on alternative algebras. The absence of a publication by Artin suggests Zorn credited Artin with the theorem because of a private communication between student and advisor.

Follow up after comments on the relation between Artin's theorem and the Artin-Zorn theorem: The theorem which Zorn attributes to Artin, that in an alternative ring the subring generated by any two elements is associative, is not limited to finite rings, only the corollary, now known as the Artin-Zorn theorem is. I quote from Zorn's 1930 paper:

Herr Artin, der diese Arbeit veranlat hat, hat auch einen Teil derErgebnisse, wie die Existenz des Einheitselements und dieReduzibilitt in einfache Systeme unter der Annahme, das dashalbeinfache System ber einem Grundkrper der Charakteristik Nullendlich ist, bewiesen, ferner den schnen Satz, da ein aus zweiElementen erzeugtes alternatives System assoziativ ist, mit derinteressanten Folgerung: Ein nullteilerfreies alternatives System mitendlich vielen Elementen ist Galoisfeld. Darber hinaus bin ich ihmfr viele Hinweise und Ratschlge zu Dank verpflichtet.

I am an undergrad senior math major taking a gap year looking to become an actuary. However, I still want to continue learning pure math. I've been looking for a relatively high level text to self study for the next few months. I'm already a good chunk through the first chapter of Lang's "Algebra" and everything is flowing nicely. However, I have not done any of the exercises. Also, there are a lot of topological examples and things not purely algebraic which I would need to review (I have access to Munkres). In the end, I want to learn a lot of math (which could possibly help with my thesis) and solve a lot of problems. For background, my algebra class sophomore year used Artin's Algebra and I have also taken a seminar on algebraic combinatorics. Are there other texts (within or outside abstract algebra) better-suited for what I'm looking for? And finally, is the Companion to Lang's a supplement/fleshing out of the material, or more of a guide to getting at the solutions?

In my opinion, Lang's Algebra has an excellent choice of topics for someone who wants to do further work in algebraic number theory or algebraic geometry. I'm not sure whether I'd recommend it for self-study, however. My feeling is that the exposition, and the exercises, are rather uneven. If you're studying from it on your own, I think you could easily get stuck at various points where it's Lang's fault and not yours. But that's just my experience, and your mileage may vary.

Since you mentioned algebraic combinatorics, I feel obliged to mention Richard Stanley's Enumerative Combinatorics. This contains a ton of excellent exercises with estimated difficulty ratings and solutions, so I think it's perfect for self-study. Algebraic combinatorics is my field, so I'm biased, but I'd recommend Stanley over Lang for your purposes.

Algebra, Second Edition, by Michael Artin, provides comprehensive coverage at the level of an honors-undergraduate or introductory-graduate course. The second edition of this classic text incorporates twenty years of feedback plus the author's own teaching experience. This book discusses concrete topics of algebra in greater detail than others, preparing readers for the more abstract concepts; linear algebra is tightly integrated throughout.

Prerequisites for reading the book are a standard first-year graduate course in algebra (including some Galois theory) and elementary notions of point set topology. With many examples, this book can be used by graduate students and all mathematicians learning number theory and related areas of algebraic geometry of curves.

The exposition is (as usual with Artin) quite elegant, and the parallel treatment of number fields and function can be illuminating as well as efficient ... a master of the subject ... It is a true classic in the field.

Now, after another forty years, and being out of print for the last decades, Artin's classic of timeless beauty has been made available again for new generations of students, teachers, researchers, mathematics historians, and bibliophiles, very much to the benefit of the mathematical community as a whole.

I'm searching for an apt textbook(s) on Abstract Algebra for a very advanced undergraduate/graduate level course in Algebra, and would be grateful for any help. I've thought of the aforementioned texts, but additional suggestions would be welcome too.

Lastly, is it better to do any one of these books from cover to cover? Or is it better to do individual sections from each book, or perhaps one book followed by another? If it is the latter two, then could the relevant chapters/order please be pointed out to me?

Search [mathematics.se] using the (reference-request) tag, and the (abstract-algebra) tag. There have been a number of posts, from students at various levels of study in abstract algebra, seeking text recommendations. One such post, mentions Lang's Algebra. It's a very thorough text (scan the Table of Contents here, but doesn't cover category theory explicitly.)

Another [mathematics.se] post of possible interest to you is Is Serge Lang's Algebra still worth reading?. Jacobson's texts are both also recommended, as is Lang's text, in that post, as are other texts you haven't mentioned.

In addition, see a question on [mathematics.se] which addresses MacLane and Birkoffs' Algebra and it's suitability as a text for upper-level undergrads and/or graduate students. (There are additional posts on [mathematics.se] about this text. This text seems to include more of the items listed in your syllabus, than the other options.

I think all three choices you list are fine choices. I'd suggest browsing through each, to discern which style, approach(es) best suit(s) your needs. My decisions, when it comes to self-study, inevitably involves the use of two highly recommended texts. Inevitably, one of them emerges as by "basis text", but when struggling with a particular topic, the second text is very useful to have. I just would not recommend juggling too many texts at one time.

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