Existence of Split Property in Quaternion Algebra Over Composite of Quadratic Fields
Quaternions are extensions of complex numbers that are four-dimensional objects. Quaternion consists of one real number and three complex numbers, commonly denoted by the standard vectors and . Quaternion algebra over the field is an algebra in which the multiplication between standard vectors is n...
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Mathematics Department UIN Maulana Malik Ibrahim Malang
2023-11-01
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| Series: | Cauchy: Jurnal Matematika Murni dan Aplikasi |
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| Online Access: | https://ejournal.uin-malang.ac.id/index.php/Math/article/view/22881 |
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| author | Muhammad Faldiyan Ema Carnia Asep Kuswandi Supriatna |
| author_facet | Muhammad Faldiyan Ema Carnia Asep Kuswandi Supriatna |
| author_sort | Muhammad Faldiyan |
| collection | DOAJ |
| description | Quaternions are extensions of complex numbers that are four-dimensional objects. Quaternion consists of one real number and three complex numbers, commonly denoted by the standard vectors and . Quaternion algebra over the field is an algebra in which the multiplication between standard vectors is non-commutative and the multiplication of standard vector with itself is a member of the field. The field considered in this study is the quadratic field and its extensions are biquadratic and composite. There have been many studies done to show the existence of split properties in quaternion algebras over quadratic fields. The purpose of this research is to prove a theorem about the existence of split properties on three field structures, namely quaternion algebras over quadratic fields, biquadratic fields, and composite of quadratic fields. We propose two theorems about biquadratic fields and composite of quadratic fields refer to theorems about the properties of the split on quadratic fields. The result of this research is a theorem proof of three theorems with different field structures that shows the different conditions of the three field structures. The conclusion is that the split property on quaternion algebras over fields exists if certain conditions can be met. |
| format | Article |
| id | doaj-art-cfe927e2ea0d43bc8878c011d2042c64 |
| institution | Kabale University |
| issn | 2086-0382 2477-3344 |
| language | English |
| publishDate | 2023-11-01 |
| publisher | Mathematics Department UIN Maulana Malik Ibrahim Malang |
| record_format | Article |
| series | Cauchy: Jurnal Matematika Murni dan Aplikasi |
| spelling | doaj-art-cfe927e2ea0d43bc8878c011d2042c642025-08-20T03:48:30ZengMathematics Department UIN Maulana Malik Ibrahim MalangCauchy: Jurnal Matematika Murni dan Aplikasi2086-03822477-33442023-11-018214215610.18860/ca.v8i2.228817401Existence of Split Property in Quaternion Algebra Over Composite of Quadratic FieldsMuhammad Faldiyan0Ema Carnia1Asep Kuswandi Supriatna2Padjadjaran UniversityPadjadjaran UniversityPadjadjaran UniversityQuaternions are extensions of complex numbers that are four-dimensional objects. Quaternion consists of one real number and three complex numbers, commonly denoted by the standard vectors and . Quaternion algebra over the field is an algebra in which the multiplication between standard vectors is non-commutative and the multiplication of standard vector with itself is a member of the field. The field considered in this study is the quadratic field and its extensions are biquadratic and composite. There have been many studies done to show the existence of split properties in quaternion algebras over quadratic fields. The purpose of this research is to prove a theorem about the existence of split properties on three field structures, namely quaternion algebras over quadratic fields, biquadratic fields, and composite of quadratic fields. We propose two theorems about biquadratic fields and composite of quadratic fields refer to theorems about the properties of the split on quadratic fields. The result of this research is a theorem proof of three theorems with different field structures that shows the different conditions of the three field structures. The conclusion is that the split property on quaternion algebras over fields exists if certain conditions can be met.https://ejournal.uin-malang.ac.id/index.php/Math/article/view/22881quaternion algebraquadratic fieldbiquadratic fieldcompositesplit |
| spellingShingle | Muhammad Faldiyan Ema Carnia Asep Kuswandi Supriatna Existence of Split Property in Quaternion Algebra Over Composite of Quadratic Fields Cauchy: Jurnal Matematika Murni dan Aplikasi quaternion algebra quadratic field biquadratic field composite split |
| title | Existence of Split Property in Quaternion Algebra Over Composite of Quadratic Fields |
| title_full | Existence of Split Property in Quaternion Algebra Over Composite of Quadratic Fields |
| title_fullStr | Existence of Split Property in Quaternion Algebra Over Composite of Quadratic Fields |
| title_full_unstemmed | Existence of Split Property in Quaternion Algebra Over Composite of Quadratic Fields |
| title_short | Existence of Split Property in Quaternion Algebra Over Composite of Quadratic Fields |
| title_sort | existence of split property in quaternion algebra over composite of quadratic fields |
| topic | quaternion algebra quadratic field biquadratic field composite split |
| url | https://ejournal.uin-malang.ac.id/index.php/Math/article/view/22881 |
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