| Summary: | A triplet <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></semantics></math></inline-formula> of positive integers is said to be product-feasible if there exist algebraic numbers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> of degrees (over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">Q</mi></semantics></math></inline-formula>) <i>a</i>, <i>b</i> and <i>c</i>, respectively, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mi>β</mi><mi>γ</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. This work extends the investigation of product-feasible triplets started by Drungilas, Dubickas and Smyth. More precisely, for all but five positive integer triplets <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>≤</mo><mi>b</mi><mo>≤</mo><mi>c</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>≤</mo><mn>7</mn></mrow></semantics></math></inline-formula>, we decide whether it is product-feasible. Moreover, in the Appendix we give an infinite family or irreducible compositum-feasible triplets and propose a problem to find all such triplets.
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