Monthly, May 2005
Gauss's Lemma for Number Fields
by Arturo Magidin and David McKinnon
Can every polynomial with integer coefficients be factored into linear terms, each with algebraic integer coefficients? And how is this related to Gauss’s lemma and to unique factorization? Unique factorization had long been taken for granted, until Gauss explicitly noted that it should be proved. It was the failure of unique factorization in some natural settings that led to the development of algebraic number theory and, through it, of ring theory as well. We will take an excursion through some of this history, name dropping mathematicians from Euclid through Gauss and Euler, to Kummer, Kronecker, and Dedekind. Along the way we will discover how wrong proofs of Fermat's Last Theorem led in part to these developments, why ideals are called ideals, and eventually two different ways of reaching the answer to our first question. We will even glimpse, far off in the distance, quadratic reciprocity and elliptic curves.
The Fundamental Group of the Circle is Trivial
by Florian Deloup
The title does not refer to an actual refutation of a basic theorem of algebraic topology but rather is an invitation to understand what "fundamental" means in three selected examples in topology at the undergraduate level. Is our intuition right about notions such as homotopy or isotopy? To find out, experimenting on a few unorthodox definitions helps.
The Convergence of Difference Boxes
by Antonio Behn, Christopher Kribs-Zaleta, and Vadim Ponomarenko
We consider an elementary mathematical puzzle known as a "difference box" in terms of a discrete map from R4 to R4 or, canonically, from a subset of the first quadrant of R2 into itself. We identify the map’s unique canonical fixed point and answer more generally the question of how many iterations a given "difference box" takes to reach zero. (The number is finite except for boxes corresponding to the fixed point.)
Do Normal Subgroups Have Straight Tails?
by Paul E. Becker
In abstract algebra courses, students are encouraged to view groups and subgroups as objects. Such courses, however, may not answer the big questions: What do these objects look like? Do direct products and semidirect products look different? Do normal subgroups look normal? In this article, we show that matrix representations permit simple, meaningful answers to the preceding questions. Along the way, we prove an extension of Cayley’s theorem, define blocks and tails for the resulting representations, and investigate the title question: Do normal subgroups have straight tails?
Avoiding Eigenvalues in Computing Matrix Powers
by Raghib Abu-Saris and Wajdi Ahmad
Exact Solutions for Minimax Optimization Problems
by Francesc Comellas and J. Luis A. Yebra
A Simple Proof of the Gauss-Winckler Inequality
by F. G. Avkhadiev
Copositive Symmetric Cubic Forms
by Mowaffaq Hajja
Problems and Solutions
By Ronald Solomon
Reviewed by Daniel J. Madden
Integers, Polynomials and Rings
By Ronald S. Irving
Reviewed by Daniel J. Madden
Introduction to the Mathematics of Medical Imaging
By Charles L. Epstein
Reviewed by John Sylvester