  The paper aims to apply the trigintaduonion spaces to explore the physical
properties of four interactions simultaneously, including the electromagnetic
fields, gravitational fields, weak nuclear fields, and strong nuclear fields.
J. C. Maxwell first applied the algebra of quaternions to study the physical
properties of electromagnetic fields. It inspired some subsequent scholars to
introduce the quaternions, octonions, sedenions, and trigintaduonions to
research the electromagnetic fields, gravitational fields, weak nuclear fields,
strong nuclear fields, quantum mechanics, gauge fields, and curved spaces and
so forth. The algebra of trigintaduonions is able to discuss the physical
quantities of four interactions, including the field potential, field strength,
field source, linear momentum, angular momentum, torque, and force. In the
field theories described with the algebra of trigintaduonions, the weak nuclear
field is composed of three types of fundamental fields. These three fundamental
fields, related to weak nuclear fields, can describe the physical properties of
weak nuclear fields collectively. This is consistent with the conclusion of the
electroweak theory. Meanwhile the strong nuclear field consists of three types
of fundamental fields. These three fundamental fields relevant to strong
nuclear fields may investigate the physical properties of strong nuclear fields
mutually. It is coincident with the deduction of quark theory. According to the
properties of trigintaduonions, one can deduce the Yang-Mills equation related
to the gauge fields. It means that the electromagnetic field occupies a
quaternion space. The gravitational field owns one different quaternion space.
The weak nuclear fields occupy three mutually independent quaternion spaces.
The properties of weak nuclear fields are different from those of
electromagnetic fields or gravitational fields.
