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Let B and B be disjoint separable algebraic extensions of a field F, and let B = B B be their composite. Let α be an element of B and α be an element of B. Suppose α and α are primitive (resp. normal, resp. subnormal). We investigate the question of when α + α and α α are necessarily primitive (resp. normal, resp. subnormal) elements of B. (A normal element of a Galois extension is defined to be one that is part of a normal basis, and a subnormal element is defined analogously for a non-Galois extension). [ABSTRACT FROM AUTHOR]