Alladi Ramakrishnan Hall

On integrally closed domains and their applications in Number Theory.

S. K. Khanduja

Indian Institute of Science Education and Research, Mohali, India.

Let R be an integrally closed domain with quotient field K and θ be an
element of an integral domain containing R with θ integral over R and F (x) be
the minimal polynomial of θ over K. Kummer proved that if R[θ] is an integrally
closed domain then the maximal ideals of R[θ] which lie over a maximal ideal p
of R can be explicitly determined from the irreducible factors of F (x) modulo
p. We shall discuss a necessary and sufficient criterion to be satisfied by F (x) so
that R[θ] is integrally closed when R is a valuation ring. We shall also give some
applications of this criterion for algebraic number fields and derive necessary
and sufficient conditions involving only the primes dividing a, b, m, n for Z[θ] to
be integrally closed when θ is a root of an irreducible trinomial x n + ax m + b
with integer coefficients. For any pair of algebraic number fields K 1 , K 2 linearly
disjoint over K 1 ∩ K 2 , we shall show that the relative discriminants of K 1 /K
and K 2 /K to be coprime if and only if the composite ring A K 1 A K 2 is integrally
closed, A K i being the ring of algebraic integers of K i . This provides converse of
a well known result in algebraic number theory and will be discussed in a more
general setting.