Square function

In abstract algebra and number theory

In the language of quadratic forms, this equality says that the square function is a "form permitting composition". In fact, the square function is the foundation upon which other quadratic forms are constructed which also permit composition. The procedure was introduced by L. E. Dickson to produce the octonions out of quaternions by doubling. The doubling method was formalized by A. A. Albert who started with the real numberfield${\displaystyle \mathbb {R} }$ and the square function, doubling it to obtain the complex number field with quadratic form x + y, and then doubling again to obtain quaternions. The doubling procedure is called the Cayley–Dickson construction, and has been generalized to form algebras of dimension 2 over a field F with involution.