#### 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.

Done