Die ganzen zahlen hat Gott gemacht

After a long stint (1981-1999) at the Tata Institute of Fundamental Research in Mumbai, the author moved to the Indian Statistical Institute, Bangalore due to his interest in under-graduate teaching.

A quote attributed to the famous mathematician L Kronecker is `Die Ganzen Zahlen hat Gott gemacht, alles andere ist Menschenwerk.' A translation might be `God gave us integers and all else is man's work.' All of us are familiar already from middle school with the similarities between the set of integers and the set of all polynomials in one variable. A paradigm of this is the Euclidean (division) algorithm. However, it requires an astute observer to notice that one has to deal with polynomials with real or rational coefficients rather than just integer coefficients for a strict analogy. There are also some apparent dissimilarities -- for instance, there is no notion among integers corresponding to the derivative of a polynomial. In this discussion, we shall consider polynomials with integer coefficients. Of course a complete study of this encompasses the whole subject of algebraic number theory, one might say. For the most of this article (in fact, with the exception of 1.9, 2.3, 2.4 and 4.3), we adhere to fairly elementary methods and address a number of rather natural questions. To give a prelude, one such question might be ``if an integral polynomial takes only values which are perfect squares, then must it be the square of a polynomial ?"

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Address for Correspondence
B Sury 
Statistics & Mathematics Unit 
Indian Statistical Institute 
Bangalore 560 059, India. 
Email: sury@isibang.ac.in