Discriminants in Algebra and Arithmetic

Discriminants in Algebra and Arithmetic


Sudhir R. Ghorpade 1

Department of Mathematics
Indian Institute of Technology, Bombay
Powai, Mumbai 400076 India
E-mail: srg@math.iitb.ernet.in


This is an expository article based on a lecture delivered at S. P. College, Pune on February 19, 2000. We begin with the familiar notion of the discriminant of a quadratic. First, we motivate and explain how this notion can be extended to polynomials of arbitrary degree in one variable. Next. we recall the notion of discriminant in the context of field extensions, using the trace map, and prove some of its basic properties. We, then, relate the two seemingly disparate notions of discriminant. Lastly, starting from the factorization of nonzero integers as a product of primes, we discuss some basic ideas of Algebraic Number Theory, and in particular, the ramification of primes when extended to rings of integers of number fields. The notion of the discriminant of a number field is discussed and its relation with the phenomenon of ramification is described. The article ends with a number of brief remarks and pointers to literature concerning the extensions and generalizations of the notion of discriminant to ``higher dimensions'', connections with Topology, and the role of discriminant in the study of binary quadratic forms.


1 Discriminant in High School Algebra 1
2 Discriminant in College Algebra 7
3 Discriminant in Arithmetic 11
Acknowledgements 19
References 19

1 Partially supported by a `Career Award' grant from AICTE, New Delhi and an IRCC grant from IIT Bombay.

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