Degree of the Product of Two Algebraic Numbers One of Which Is of Prime Degree
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"...
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-03-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/11/6/1485 |
| _version_ | 1797610297861603328 |
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| author | Paulius Virbalas |
| author_facet | Paulius Virbalas |
| author_sort | Paulius Virbalas |
| collection | DOAJ |
| description | 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>. |
| first_indexed | 2024-03-11T06:13:31Z |
| format | Article |
| id | doaj.art-90ba6c6f2a1f4f7daf0a53a4f86d10dc |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T06:13:31Z |
| publishDate | 2023-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-90ba6c6f2a1f4f7daf0a53a4f86d10dc2023-11-17T12:29:18ZengMDPI AGMathematics2227-73902023-03-01116148510.3390/math11061485Degree of the Product of Two Algebraic Numbers One of Which Is of Prime DegreePaulius Virbalas0Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, LithuaniaLet <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>.https://www.mdpi.com/2227-7390/11/6/1485degree of an algebraic numberGalois theorytransitive permutation groups |
| spellingShingle | Paulius Virbalas Degree of the Product of Two Algebraic Numbers One of Which Is of Prime Degree Mathematics degree of an algebraic number Galois theory transitive permutation groups |
| title | Degree of the Product of Two Algebraic Numbers One of Which Is of Prime Degree |
| title_full | Degree of the Product of Two Algebraic Numbers One of Which Is of Prime Degree |
| title_fullStr | Degree of the Product of Two Algebraic Numbers One of Which Is of Prime Degree |
| title_full_unstemmed | Degree of the Product of Two Algebraic Numbers One of Which Is of Prime Degree |
| title_short | Degree of the Product of Two Algebraic Numbers One of Which Is of Prime Degree |
| title_sort | degree of the product of two algebraic numbers one of which is of prime degree |
| topic | degree of an algebraic number Galois theory transitive permutation groups |
| url | https://www.mdpi.com/2227-7390/11/6/1485 |
| work_keys_str_mv | AT pauliusvirbalas degreeoftheproductoftwoalgebraicnumbersoneofwhichisofprimedegree |