School of Mathematics Seminars and Lectures

`On splitting of primes and simple extensions of integrally closed domains'

by Prof. S.K. Khanduja (IISER, Mohali)

Thursday, November 27, 2014 from to (Asia/Kolkata)
at AG-77
Description We will discuss an extension of the  classical  Dedekind's  theorem
regarding
splitting of rational primes in algebraic number fields  as well as of its
converse when the base field is a valued field of arbitrary rank.   Let $R$
be
the valuation ring of a Krull valuation  defined on a field $K$ and $S$ be the integral closure of $R$ in a finite extension $L$ of $K.$ A set of conditions will be described which are necessary as well as sufficient for $S$ to be a simple  ring extension of $R,$ i.e., $S=R[\theta]$ for some $\theta.$ The well  known theorem  of Dedekind characterizing those rational primes $p$ which divide the index of an algebraic number field will be deduced. Some related open problems will also be mentioned.