Book Description
The Fundamental Theorem of Algebra states that any complex polynomial must have a complex root. This basic result, whose first accepted proof was given by Gauss, lies really at the intersection of the theory of numbers and the theory of equations, and arises also in many other areas of mathematics. The purpose of this book is to examine three pairs of proofs of the theorem from three different areas of mathematics: abstract algebra, complex analysis and topology. The first proof in each pair is fairly straightforward and depends only on what could be considered elementary mathematics. However, each of these first proofs lends itself to generalizations, which in turn, lead to more general results from which the fundamental theorem can be deduced as a direct consequence. These general results constitute the second prooof in each pair. To arrive at each of the proofs, enough of the general theory of each relevant area is developed to understand the proof. In addition to the proofs and techniques themselves, many applications such as the insolvability of the quintic and the trascendence of e and pi are presented. Finally, a series of appendices give six additional proofs including a version of Gauss' original first proof. The book is intended for junior/senior level undergraduate mathematics students or first year graduate students. It is ideal for a "capstone" course in mathematics. It could also be used as an alternative approach to an undergraduate abstract algebra course. Finally, because of the breadth of topics it covers it would also be ideal for a graduate course for mathmatics teachers. ... Read more

Customer Reviews (2)

Too Poorly Written
To be sure, the aim of this book is interesting, but it's too poorly written. There are so many serious mistakes that I wonder why the students who took the course of these authors didn't point them out.
For example, the latter half of Lemma 4.1.5 is wrong for it lacks (-i), and Theorem 7.3.3 is wrong for |K:F| is not equal to the number of automorphisms of K fixing F, but is equal to the number of injective isomorphisms form K to algebraic closure of F fixing F.
I recommend you to read more reliable books.

A wonderful text!
This text gives a number of proofs of the Fundamental Theorem of Algebra (FTA).The proofs come from diverse areas of mathematics including complex analysis, abstract algebra (Galois theory) and topology.A version ofGauss's original proof is even given.The background information leadingup to each proof makes the book readable for a math major who has taken afirst course in abstract algebra.A student who uses this text will haveseen a lot of really interesting mathematics by the end and will have asense of how the mathematics can be used. ... Read more

Book Description
There are a number of very good books available on linearalgebra. From this one might deduce that the existing bookscontain all that one needs to know in the best possible form andthat any new book would just repeat material in the old ones.However, new results in linear algebra appear constantly, as donew, simpler, and better proofs of old results. Many linearalgebra results obtained in the past thirty years are accessibleto undergraduate mathematics majors, but are usually ignored bytextbooks. In addition, more than a few interesting old resultsare not covered in many books. In this book, Prasolov providesthe basics of linear algebra, with an emphasis on new resultsand on nonstandard and interesting proofs. The book featuresabout 230 problems with complete solutions. It would be a finesupplementary text for an undergraduate or graduate algebracourse. ... Read more

Customer Reviews (1)

A good book on finite dimentional linear algebra
The book opens with determinants ... excepted for this point, I think this book is worth the money it costs.

All you need to know in linear algebra at undergraduate level is here. ... Read more

Book Description
Various types of approximation theorems are frequently used in general commutative algebra, and they have been found to be useful tools in valuation theory, the theory of Abelian lattice ordered groups, multiplicative ideal theory, etc. Part 1 of this volume is devoted to the investigation of approximation theorems from a classical point of view. The chapters of this part deal with fields and rings, partly ordered groups, and with multirings and d-groups. Part II investigates approximation theorems from a general, categorical point of view. This part is essentially self-contained and requires only a basic knowledge of category theory and first-order logic. For researchers and graduate students of commutative algebra, category theory, as well as applications of logic. ... Read more

Book Description

The unpublished writings of Helmut Hasse, consisting of letters, manuscripts and other papers, are kept at the Handschriftenabteilung of the University Library at Göttingen. Hasse had an extensive correspondence; he liked to exchange mathematical ideas, results and methods freely with his colleagues. There are more than 8000 documents preserved. Although not all of them are of equal mathematical interest, searching through this treasure can help us to assess the development of Number Theory through the 1920s and 1930s.