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Can anyone recommend a good reference for brushing up on quadratic forms?

They keep coming up (quite naturally of course) in the context of differential geometry and I find I am rustier than I remembered. I've looked through a large number of linear algebra books and found mostly short sections that cover the basics: reduction to canonical form and Sylvester's criterion, but nothing that goes much beyond. I've also found others that get quite deep into the issue from a more abstract algebraic point of view, but they seemed like they would require a significant detour from my current studies. Does anyone know any books that provide something in between?

(Something available as a pdf that I can download/buy online would be perfect, but any other reference would help greatly as well).

Thanks in advance

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I hope you don't mind, but I have changed the title to make it a little clearer. –  Old John Dec 31 '12 at 14:04
    
Usually Pfister's Quadratic Forms with Applications to Algebraic Geometry and Topology is a good start. You can also refer Lam's Introduction to Quadratic Forms over Fields, American Mathematical Society. –  Ram Dec 31 '12 at 14:28
    
What exactly do you have in mind when you say "nothing that goes much beyond"? What other results do you want to have/encounter in your study of geometry that needs more than canonical form? –  user27126 Dec 31 '12 at 14:43
    
@OldJohn: thank you –  Amos Joshua Jan 1 '13 at 22:24
    
@Sanchez: I'm not sure, I don't know what I don't know. The books I found (I truly looked through quite a few) did not seem like the best choices, it occurred to me that asking the community for suggestions is probably better than randomly trying books. –  Amos Joshua Jan 1 '13 at 22:36
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1 Answer

up vote 2 down vote accepted

Here are three PDFs that I became aware of during my own searches on this question some time ago:

http://www.math.jussieu.fr/~karpenko/publ/Kniga.pdf

http://www.math.miami.edu/~armstrong/685fa12/pete_clark.pdf

http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/bilinearform.pdf

I think a "middle" text that is both affordable and easy to obtain is Jacobson's Basic Algebra I Chapter 6 on metric vector spaces.

T.Y. Lam's Introduction to quadratic forms also sounds like a good bet. I say that because I haven't read this book yet, but judging from the author's other books, I bet this one must be good too. Good luck!

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Lam's book is very good, but it's prerequisites requires some what advanced knowledge, I don't think it serves as nice introduction to the subject.. –  Ram Dec 31 '12 at 17:21
    
Perfect - thank you! I especially like pete_clark.pdf –  Amos Joshua Jan 1 '13 at 22:24
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