ON THE ISOMORPHISM CONDITION AMONG SUBFIELDS OF THE ALGEBRAIC CLOSURE OF Γ(p)

Abstract

Let  and  be Steinitz numbers. Let  be a prime and  a field of  elements. The algebraic closure  is the union of all fields  for positive integers . Theorem 9.8.4 in (Roman, 2005) states that  if and only if the two Steinitz numbers  and  are equal. In this paper, we continue to develop the above result by showing that the different subfields within  have distinct field structures. Specifically, it is proven that a field isomorphism between  and  exists if and only if . This result provides an important characterization of the subfield structure within .