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

## Abstract

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.

Contents

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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