lõppkvaternioonid

Lõppkvaternioonid, also known as finite quaternions, are a generalization of the concept of quaternions to a finite field. Quaternions themselves are a number system that extends complex numbers, with four components and a non-commutative multiplication. Lõppkvaternioonid adapt this structure to operate within the arithmetic of a finite field, such as GF(p) for a prime p.

The definition of lõppkvaternioonid typically involves elements of the form a + bi + cj + dk, where a, b, c, and d belong to a finite field, and i, j, and k are basis elements satisfying specific multiplication rules similar to those of regular quaternions, but adapted for the finite field's modular arithmetic. These rules usually include i^2 = j^2 = k^2 = ijk = -1 (modulo p). The specific field and the chosen values for the basis element squares can lead to different types of lõppkvaternioonid.

Lõppkvaternioonid have found applications in various areas of mathematics and computer science. They are used in coding theory, particularly in the construction of error-correcting codes. Their algebraic structure also makes them relevant in cryptography, where they can be employed in the design of secure algorithms. Furthermore, lõppkvaternioonid appear in the study of finite geometries and group theory. Research in this area explores their properties, such as their division properties and their relationship to other algebraic structures over finite fields. The study of lõppkvaternioonid is an active area of abstract algebra.