| Summary: | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> be two algebraic numbers such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">deg</mo><mo>(</mo><mi>α</mi><mo>)</mo><mo>=</mo><mi>m</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">deg</mo><mo>(</mo><mi>β</mi><mo>)</mo><mo>=</mo><mi>p</mi></mrow></semantics></math></inline-formula>, where <i>p</i> is a prime number not dividing <i>m</i>. This research is focused on the following two objectives: to discover new conditions under which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">deg</mo><mo>(</mo><mi>α</mi><mi>β</mi><mo>)</mo><mo>=</mo><mi>m</mi><mi>p</mi></mrow></semantics></math></inline-formula>; to determine the complete list of values <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">deg</mo><mo>(</mo><mi>α</mi><mi>β</mi><mo>)</mo></mrow></semantics></math></inline-formula> can take. With respect to the first question, we find that if the minimal polynomial of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> 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> is neither <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mi>p</mi></msup><mo>+</mo><mi>c</mi></mrow></semantics></math></inline-formula> nor <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mi>x</mi><mo>+</mo><msup><mi>c</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula>, then necessarily <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">deg</mo><mo>(</mo><mi>α</mi><mi>β</mi><mo>)</mo><mo>=</mo><mi>m</mi><mi>p</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mi>β</mi></mrow></semantics></math></inline-formula> is a primitive element of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">Q</mi><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></mrow></semantics></math></inline-formula>. This supplements some earlier results by Weintraub. With respect to the second question, we determine that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>></mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> divides <i>m</i>, then for every divisor <i>k</i> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>, there exist <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">deg</mo><mo>(</mo><mi>α</mi><mi>β</mi><mo>)</mo><mo>=</mo><mi>m</mi><mi>p</mi><mo>/</mo><mi>k</mi></mrow></semantics></math></inline-formula>.
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