Existence of Split Property in Quaternion Algebra Over Composite of Quadratic Fields

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INTRODUCTION
Quaternions are a useful tool for comprehending a variety of physics and kinematic concepts.Quaternion are frequently employed to address optimization issues involving predicting rigid body transformations, particularly in the disciplines of computer vision, computer graphics, and animation [1].Quaternions are extensions of complex numbers.Complex numbers denoted by ℂ are a subset of quaternion numbers with the notation ℍ. Quaternion numbers are also called hypercomplex numbers because they are a generalization of the complex number system.Quaternion numbers are applied in various applications such as color image filtering, segmentation, and 3-dimensional impulse response filter design [2].Quaternion numbers have four components, namely one real number and three imaginary numbers that have the form:  =  +  +  + , where , , , and  are real numbers and , , and  are imaginary numbers [3].
Quaternion algebra is a generalization of vector algebra.Quaternion algebra was first presented by Hamilton more than a hundred years ago but has only been practically applied recently, especially in the industrial field.A four-dimensional vector can be used to symbolize quaternion algebra [4].A vector space  over a field  with a bilinear mapping makes up a quaternion algebra.An algebra  is said to be a division algebra if for every ,  ∈ , with  •  = 0 implies  = 0 or  = 0.In other words, there are no zero divisors in division algebra.Any field  can be used to generalize quaternion algebra in place of the real number field .A quaternion algebraic system is a non-commutative algebraic system defined the multiplication of its imaginary vectors, i.e.,  2 = ,  2 = ,  2 = −, and  = − = , where ,  ∈  [5].
A quadratic field is a two-degree field over a rational number ℚ.A quadratic field has the form  + √, where ,  ∈ ℚ and  is a square-free integer.An integer that has no repeating elements in its prime decomposition is said to be square-free.The symbol for a quadratic field is ℚ(√).A biquadratic field is a quadratic field that contains two different square-free integer elements.Suppose ℚ is a rational number and ,  are distinct square-free integers; the field formed by √ and √ to ℚ is denoted by ℚ(√, √).The notation ℚ(√, √) can be defined as a quadratic field over rational numbers ℚ if ℚ(√, √) = ℚ(√ + √) and √ + √ has unique minimal polynomial  4 − 2( + ) 2 + ( + ) 2 .The term "integer of the field ℚ(√, √)" refers to any element in ℚ(√, √) that has a monic equation of degree ≥ 1 with rational integral coefficients [6].A composite number is a natural number greater than one and is the inverse of a prime number.The composite of  quadratic fields is the quadratic field of number  and is denoted by ℚ(√ 1 , √ 2 , … , √  ) [7].
A lot of research has been done to study quaternion algebras over a field.Research [8] discusses the characteristics of quaternion numbers.This research studies the properties of quaternion numbers in general but has not studied them further in the field.Research [9] discusses the split quaternion algebra over a certain field.This research further studies the split properties of quaternion algebras over certain fields.The second research [10] discusses the class of quaternion algebra in the field.This research studies the special properties of division algebra in quaternion algebra.The characteristics of quaternion algebras over quadratic fields are covered in research [11].This study explores the properties of quaternion algebras over quadratic fields as well as the prerequisites and requirements for using quaternion algebras over quadratic fields as division algebras.Research [7] discusses the development of previous research, namely the characteristics of the quaternion algebra over the composite of  quadratic fields.This research reveals the characteristics of quaternion algebra and its application to Fibonacci numbers.Based on this research, a study is formed that discusses the characteristics of the quaternion algebra over the composite of  quadratic fields in a division algebra.
From these studies, we found a research gap that needs to be developed further.We will develop a theorem on split existence for a wider field, namely quaternion algebra over the composite of  quadratic fields.The purpose of this research is to prove the theorem about split existence on three algebraic structures over the field, namely a quadratic field, a biquadratic field, and a composite of  quadratic field.Since split properties on quaternion algebras over quadratic fields have already been theorized, this study will focus on split properties on quaternion algebras over biquadratic fields and composites of  quadratic fields, two areas of unexplored research.A theorem on the split properties of quaternion algebras over quadratic fields was described by Acciaro and Savin in 2018 [12].Two theorems regarding the split properties on biquadratic fields and a composite of  quadratic fields that are related to theorems about the split properties on quadratic fields are proposed.The results of this study will show the existence of split properties that always exist in quaternion algebras under certain conditions.In addition, the results of this study show the different conditions required for quaternion algebras to be split.Larger fields tend to require more conditions for the split existence theorem on quaternion algebras to hold.This research can be useful for the advancement of algebraic science studies, especially in the field of algebra over the field.

METHODS
This section describes the definitions, theorems, lemmas, and propositions used in this research to answer the main research problem, namely answering the characteristics through theorem proving of the quaternion algebra over the composite of  quadratic fields that have split properties.The method used in answering this main problem is a literature study that focuses on the material of quaternion algebra over the composite of  quadratic fields and its properties.First, we will prove the theorem about the sufficient condition of quaternion algebra over quadratic fields having split properties.Then, the research will be extended by proving the theorem about the sufficient condition that the quaternion algebra over the biquadratic fields has split properties.Finally, the research will be extended again by proving the theorem about the sufficient condition of quaternion algebra over the composite of  quadratic fields that also has the split property.The following is a definition that becomes the theoretical basis for answering research problems about quaternion algebra.Definition 1. (See [13]) (Quaternion) Quaternions are an extension of complex numbers.Quaternions have number elements consisting of one real number and three imaginary numbers.Quaternions mathematically, can be written as follows: ℍ = { +  +  + ; ∀, , ,  ∈ ℝ}; where the following multiplication rule conditions apply: 1.  2 =  2 =  2 = −1; 2.  = ,  = ,  = ,  = −,  = −,  = −; 3.Each  ∈ ℝ is commutative with , , and .
Definition 3. (See [15]) (Algebra over field) Assume that  is a field and that  is a vector space over  with an addition operation in binary form from  ×  to .Vector space  is an algebra over a field  if it satisfies the following axioms: () = () = (), ∀ ∈   ∀,  ∈ .

Definition 6. (See [17]) (Norm)
Consider  =  +  +  +  to be a quaternion number.The norm of quaternion can be defined as follows:

Definition 8. (See [18]) (Field Extension)
Suppose  and  are fields.If and only if  is a subfield of , then field  is an extension of .Field  is a vector space over a field  denoted by / and its dimension is denoted by [: ].

Definition 9.
(See [19]) (Quadratic Field) The field of rational numbers ℚ of degree two extends into the quadratic field.The quadratic field has the form  + √, where  is a square-free integer represented by ℚ(√) in the form of a quadratic field.If  > 0, the field is referred to as a real quadratic field.If  < 0, it is referred to as an imaginary quadratic field.

Definition 10. (See [20]) (Square-free)
A number is said to be squarefree if its prime decomposition does not contain repeated factors.Therefore, all prime numbers are squarefree.

Definition 11. (See [11]) (Division Algebra)
Associative algebra  is a kind of algebra. over a field  is a division algebra if and only if it has a multiplication identity element of 1 ≠ 0 and a left and right multiplication inverse for each non-zero element in .If  is a finite-dimensional algebra, it is a division algebra if and only if it has no zero divisors.Definition 12. (See [16]) (Division Ring) Division algebra is an algebra over the division ring  (every nonzero element has an inverse).
Definition 13. (See [6]) (Biquadratic Field) Assume that  is an odd prime integer and that  ∈ ℤ.This is how the legendre symbol ) is defined: then  is a non-quadratic residue.
Definition 16. (See [11]) Suppose  is a prime ideal of the ring   .If  is quadratic in , then its quadratic residue symbol is Definition 17. (See [23]) (Hilbert Equation) For the field  and ,  ∈  * = \{0}.We state that is a Hilbert Equation and the solution (, , ) ∈  3 is trivial if and only if  =  =  = 0, otherwise it is nontrivial.

Definition 18. (See [23]) (Hilbert Symbol)
Suppose  is a field and  ⊆  is a subring.Define the mapping that resolves a quadratic diagonal equation in three variables with coefficients in  and is nontrivial  solvable: Proposition 19.(See [24]) (Discriminant of quadratic field) Let ∆  be the discriminant of a square-free integer  in the quadratic field  = ℚ(√).
The following conditions hold: Theorem 20. (See [6]) (Discriminant of biquadratic field) Assume that  = (, ) is a Hilbert Symbol,  =  1 , and that  =  1 , and that ( 1 ,  1 ) = 1.ℚ(√, √) has the following discriminant: Theorem 21. (See [25]) (Split Properties) Suppose  is an extension quadratic field of  and   is a quaternion algebra over .If no ramified prime of  in   is split in , then there is an insertion of  into   .
Theorem 22. (See [12]) (Prime decomposition in the quadratic field) Suppose  ≠ 0,1 is an integer with no squares (square-free integer).Suppose ∆  is the discriminant of  and   is the integer ring of the quadratic field  = ℚ(√).Assume that  is an odd prime number.Next, we have: ),  2 is a separate prime ideal in   , and 2  =  1 •  2 ; f.If and only if  ≡ 5 ( 8), the prime 2 is inert in   .

Theorem 23. (See [12]) (Split properties of prime numbers in a biquadratic field)
Suppose  1 and  2 are two distinct square-free integers not equal to one, and suppose  3 = ( 1 , 2 ) gcd( 1 , 2 ) .Suppose   is the integer ring of the quadratic field  and    is the integer ring of the quadratic subfield   = ℚ(√  ), where  = 1,2,3.Let's assume  is a prime number.When  is split in every    , where  = 1,2,3, then and only then  is split in   .
Theorem 24.(See [12]) (Split for composite field) Suppose  is a prime of ℚ that is split in every field  1 ,  2 , … ,   .Then  is split in the composite field  1  2 …   .

RESULTS AND DISCUSSION
A development of complex algebra known as quaternion algebra may be created in a number of areas of mathematics, including algebra, analysis, geometry, and arithmetic.In the field of algebra, quaternion algebra can be developed based on algebraic properties, namely split, inert, and ramified.Split quaternion algebras tend to be easier to decompose into simpler algebras.Therefore, a sufficient condition is needed so that a quaternion algebra is said to be a split algebra.First, what will be reviewed in this result and discussion is the existence of split properties on quaternion algebras over quadratic fields have been proved by Acciaro and Savin [12].Then proceed to prove the split properties for a wider field, namely the biquadratic field and the composite of  quadratic fields.The theorem on the split properties of the quaternion algebra over the quadratic field is shown as follows: Theorem 26 (See [12]) Suppose  ≠ 0,1 are square-free integers,  ≢ 1 ( 8), and ,  are prime integers, with  ≥ 3,  ≠ .Let ∆  is the discriminant of  and   is the ring of integers for the quadratic field  = ℚ(√).Then:   Furthermore, a new theorem is given that will be proved about the split properties of quaternion algebras over biquadratic fields.The biquadratic field is wider and more complicated than the quadratic field, so it requires some additional conditions on this theorem compared to the previous theorem.The theorem on the split property of a biquadratic field is a research novelty formed from Theorem 26.The establishment of this theorem is done to guarantee the existence of sufficient conditions that make a quaternion algebra over biquadratic field is split.The following is the content and proof of the theorem on the existence of split properties on quaternion algebras over biquadratic fields.

Theorem 27
Suppose  1 ,  2 ≠ 0,1 are square-free integers, where  1 ,  2 ≢ 1 ( 8) and . Suppose ,  are prime integers, with  ≥ 3,  ≠ .Suppose   is the ring of integers for the biquadratic field  = ℚ(√ 1 , √ 2 ) and that    is the ring of integers for the quadratic subfield   = ℚ(√  ), where  = 1,2,3, with discriminant ∆   .Then: a.If  ≥ 3 and legendre symbol ( The proof is the same as in the first case (for   1 ) by replacing  1 with  2 and  1 with  2 .This happens because the criteria for the reviewed square-free integers  1 and  2 are the same.
Third case for   3 : By Proposition 19, it is known that the discriminant of does not hold, and neither does  2 .It follows that  1 ≡ 1 ( 4) and  2 ≡ 1 ( 4).It is known in the theorem that 1. Consider the cases  1 = 2 and  2 = 2′ so that 2. Consider the cases  1 = 2 and  2 = ′′ so that 3. Consider the cases  1 =  and  2 = 2′ so that 4. Consider the cases  1 =  and  2 = ′′ so that From these four cases, it can be concluded that  3 is the product of two or more primes once, so that the prime decomposition is not repeated.Therefore,  3 is also a square-free integer.
In the theorem, it was mentioned that (  Quaternion algebras over quadratic fields and biquadratic fields can be split under certain conditions, as has been demonstrated in the previous theorem.After that, it will be demonstrated that splitting the quaternion algebra over the composite of  quadratic fields is adequate.The goal of this theorem's proof to establish the prerequisites for splitting the quaternion algebra over the composite of  quadratic fields.This theorem is a new statement that builds on Theorem 26 by studying the split properties of composites fields.The following is the content of the theorem and its proof of existence of split on quaternion algebra over the composite of  quadratic fields.
So, mathematical induction proves that the split property theorem holds for composite of  quadratic fields.
b.The proof is the same as that of part (a).It is only necessary to substitute  = 2 and apply Theorems 26 and 27 of part (b) to answer the proof of Theorem 28.
So, the theorem proves that quaternion algebra over composite of  quadratic field is split to be true under these conditions.Theorems 26, 27, and 28 have differences in the fields, namely the quadratic field in Theorem 26, the biquadratic field in Theorem 27, and the composite of  quadratic fields in Theorem 28.It can be seen that the wider field considered in these theorems to be split, the more additional conditions are required.Theorem 26 applies according to the conditions mentioned in the theorem.Theorem 27 adds additional conditions from theorem 26, namely  3 = ( 1 , 2 ) ( 1 , 2 ) and a quadratic subfield.Theorem 28 adds an additional condition from Theorem 26 and 27, namely, that primes  and  do not divide the square-free integer   .Thus, the split property will apply more easily to quadratic fields than to biquadratic fields and composite of  quadratic fields.

CONCLUSIONS
The conclusion of this research is that the split property of quaternion algebra will apply more easily to smaller fields.The larger the field under review, the more condition will be needed so that the quaternion algebra over the field is split.The Legendre symbol of the field determinant of prime numbers generally must not equal one in order for a quaternion algebra to be split.
[22]nition 14. (See[21]) (Discriminant) Suppose  to be a field of degree  with complex insertions  1 , … ,   and let  1 , … ,   ∈ .The discriminant ∆( 1 , … ,   ) of these -tuples is defined as the square of the determinant of an  ×  matrix.(())Definition15. (See[22]) (Legendre Symbol) and only if |∆  ,  is ramified in   .Therefore,   = (, √) then and only if  is split in   .If  1 and  2 are different prime ideals in   , then   =  1 •  2 .If and only if  ≡ 1 ( 8) exists, the prime 2 is split in   .Given that  1 = Suppose ∆ is the discriminant of the quaternion algebra  ℚ(√) (, ).A prime positive integer  ′ is known to be ramified in  ℚ(√) (, ) if  ′ |2∆, indicating that  ′ |2.Since  ≢ 1 ( 8) and the decomposition of 2 in ring   yield that 2 is not split in .Applying Theorem 21, there is no ramified prime such that  ℚ(√) (, ) is split.Based on the above proof, it can be concluded that the quaternion algebra is  is ramified or inert in   so  is not split in the quadratic field .Suppose ∆ is the discriminant of the quaternion algebra  ℚ(√) (2, ) and a prime positive integer  ′ is known to be ramified in  ℚ(√) (2, ) if  ′ |2∆, indicating that  ′ |2.Since  ≢ 1 ( 8) and the decomposition of 2 in ring   yield that 2 is not split in .Applying Theorem 21, no prime is ramified, so  ℚ(√) (2, ) is split.Based on the above proof, it can be concluded that the quaternion algebra is split with this condition.So, the sufficient condition for quaternion algebra to be split is when  ≥ 3, ( According to Definition 16,  and  are both ramified in   or inert in   , therefore  and  are not split in the quadratic field . 2 and legendre symbol ( then  ℚ(√ 1 ,√ 2 ) (2, ) is split in    .We will prove that both prime  and  split in   1 ,   2 , and   3 .First case for   1 :By Proposition 19, it is known that the discriminant of∆  1 is  1 (if  1 ≡ 1 ( 4)) or 4 1 (if  1 ≡ 2,3 ( 4)).Since in the theorem it is known that (  and  are ramified or inert in   1 so they are not split in the quadratic subfield  1 .In Theorem 22, it is known that  is ramified in   1 if and only if |∆  1 .By Lemma 25 points (a) and (c), the discriminant of the quadratic subfield  1 is ∆  1 = 2 or .Since |2 or |,  is ramified in   1 .It is also true for  which is also ramified in   1 .Since  1 ≢ 1 ( 8) and the decomposition of primes ,  imply that ,  are not split in  1 .By Theorem 21, it is clear that there are no ramified primes such that  ℚ(√ 1 ) (, ) is split.Second case for   2 : Proof:a.According to Theorem 23, a prime ′ split in   only if ′ split in    , where  = 1,2, and 3.
By Definition 16, the primes  and  are ramified or inert in   3 so they are not split in the quadratic subfield of  3 .In Theorem 22, it is known that  is ramified in   3 if and only if |∆  3 .The discriminant of the quadratic subfield of  3 is  ′ , 2 ′  ′ , or ′′.Since |′ or |2′′ or |′′ so  is ramified in   3 .The same holds for  which is also ramified in   3 .Since  1 ,  2 ≢ 1 ( 8) then  3 ≢ 1 ( 8).Since  3 ≢ 1 ( 8) and the decomposition of primes  and  imply that  and  are not split in  3 .Based on Theorem 21, it is clear that there are no ramified primes such that  ℚ(√ 3 ) (, ) is split.From these three cases, it can be concluded that the primes  and  split in the ring of integers   1 ,   2 , and   3 .Since primes  and  split in every    ( = 1,2,3), then by Theorem 23 it can be said that primes  and  split in   , where  = ℚ(√ 1 , √ 2 ).So, it is proved that quaternion algebra over biquadratic field is split.b.According to Theorem 23, a prime ′ split in   only if ′ split in    , where  = 1,2, and 3. We will prove that prime  is split in   1 ,   2 , and   3 .Based on definition 18, the prime  is ramified or inert in   1 so it is not split in the quadratic subfield  1 .In Theorem 22, it is known that  is ramified in   1 if and only if |∆  1 .By Lemma 24 point (b), the discriminant of the quadratic subfield  1 is ∆  1 = 2.Since |2, then  is ramified in   1 .Since  1 ≢ 1 ( 8) and the decomposition of prime  imply that  is not split in  1 .By Theorem 21, it is clear that there are no ramified primes such that  ℚ(√ 1 ) (2, ) is split.Second case for   2 : The proof is the same as in the first case (for   1 ) by replacing  1 with  2 and  1 with  2 .This happens because the criteria for the reviewed square-free integers  1 and  2 are the same.Third case for   3 : By Proposition 15, it is known that the discriminant of ∆  1 is  1 (if  1 ≡ 1 ( 4)) or 4 1 (if  1 ≡ 2,3 ( 4)) and the discriminant of ∆  2 is  2 (if  2 ≡ 1 ( 4)) or 4 2 (if  2 ≡ 2,3 ( 4)).Lemma 19 explains that the discriminant of a quadratic field is 2 (if  = 2 and  ≡ 3 ( 8)).From proposition 15 and Lemma 19, the relation of the discriminant of the quadratic field is  1 = 2 or 4 1 = 2 and  2 = 2 or 4 2 = 2.Since  1 ,  2 are square-free integers, the relationship  1 = By definition 18, the prime  is ramified or inert in   3 so it is not split in the quadratic subfield of  3 .In theorem 22, it is known that  is ramified in   3 if and only if |∆  3 .The discriminant of the quadratic subfield of  3 is ′.Since |′, so that  is ramified in   3 .Since  1 ,  2 ≢ 1 ( 8) then  3 ≢ 1 ( 8).Since  3 ≢ 1 ( 8) and the decomposition of prime  imply that  is not split in  3 .Based on theorem 21, it is clear that there are no ramified primes such that  ℚ(√ 3 ) (2, ) is split.From these three cases, it can be concluded that the primes 2 and  split in the ring of integers   1 ,   2 , and   3 .Since primes 2 and  split in every    ( = 1,2,3), then by Theorem 23 it can be said that primes 2 and  split in   , where  = ℚ(√ 1 , √ 2 ).So, it is proved that quaternion algebra over biquadratic field is split.