Number Theory and Modern Algebra, A Personal Approach by Franz Rothe

His experience of teaching mathematics at the university of North Carolina at Charlotte have made Prof. Dr. Franz Rothe a competent expert, who is now writing several mathematics books during his retirement. By the way, his retirement has been enforced on him because of severe eye problems. But the diligent decade long gathering of notes,  exercises and other material from courses allow him to continue mathematical work as an author of mathematical books. From his courses, especially in geometry, number theory and modern algebra, and graph theory the material is nearly overwhelming, he tells.

As a new incentive the computer language mathematica is available. Rothe tells that the direct use in class had always been too complicated and time consuming,  and distracting  in the higher and more abstract topics. Now he finally took the effort to master the necessary, but a bid strange computer language mathematica by Wolfram, good enough for the use to solve interesting problems. Rothe’s interest in geometry has taken him to investigations on geometric constructions by compass and straightedge.  

He has discovered a  bid simpler way to get to the regular $17$ gon. This problem has been solved by Gauss using  higher number theory, but Rothe has an approach based on simple trigonometric identities. One gets the entire tree of quadratic equations which have to be solved  in several steps, to finally arrive at the regular seventeen gon. But the book has also large parts which are more elementary, and where the graphing calculator may be used for illustrative computations. Even here the necessary programming code is spelled out. This is the case for Euclid’s  algorithm from antiquity for  the greatest common divisor. Another related example is the use of many remainder for the computing larger quantities, known as the Chinese remainder theorem because it was already known in Chine in the middle ages.

The book contains a long treatise on prime number, from the basics, the sieve of Erathothenes, many proofs for the existence of infinitely many prime numbers, investigations of their distribution and several tests for primes, and much more.

Naturally, there are parts of considerable sophistication. Besides the classical results, Rothe’s interest to invent mathematical problems for competitions, and pose problems for individual work with gifted students are giving more life to this work, and has lead him to new aspects which have not been seen in common algebra textbooks before. Not only general results and proofs, but challenging problems are the source of interesting mathematics.

The work has come to a stop by the pure need to gather and order the material into a whole, even long before half of the possible topics have been exploited.  So one may be looking forward to see in the future more books by the same author.

Number Theory and Modern Algebra Notes: A Personal Approach

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