On The Minimal Polynomials of the N-Generalized Quaternions
by
K. Todorov
South-West University, Blagoevgrad, Bulgaria

The quaternion algebra ha₂ is a four-dimensional algebra (over the field of real number R), with the standard basis { 1,  i,  j,  k }. We will denote the elements 1,  i,  j,  k of ha₂ by e₀,  e₁,  e₂,  e₃.

Multiplication is determined by the rules of the multiplication of the quaternion group \rt₂. The norm N(q)  =  |q| of the quaternion

q =  a0e0 + a1e1 + a2e2 + a3e3,    a0,  a1,  a2,  a3 in R.

is defined as N(q)  =  |q|  =  a₀² + a₁² + a₂² + a₃²  =  q[`q]  = [`q]q. The minimal polynomial, satisfied by the quaternion q is h(x)  =  x² -2a₀x + N(q).

The quaternion group \rt₂ can be considered also as a semigroup with the following genetic code: \rt₂ = <a, b:  a = bab  and  b = aba >.

Here we shall show a generalization \rt_n of the quaternion group \rt₂ obtain in studying a semigroup, generated by the elements of the set M_n = {a₁,   a₂,   ... ,   a_n } subject only to the relations a_i = a_ja_ia_j  for every two elements a_i, a_j in M with  j  =/=  i.

The quaternion algebra ha_n (of the quaternion group rt_n) over the field of real number R is a 2ⁿ-dimensional algebra with the standard basis { e_i, i = 0,   1,   ... ,   2ⁿ }. Multiplication is determined by the rules of the multiplication of the quaternion group rt_n.

Any quaternion q  =  \sum_i=0⁷c_if_i in rt₃ satisfies the minimal polinomial

h(x)  = x2 - 2(cof0 + c7f7)x + N(q),  where N(q)  =  3 å i=0  cif0 + 7 å i=4  cif7.

Any quaternion q  = \sum_i=0¹⁵c_if_i in \rt₄ satisfies the minimal polinomial

h(x)  = x4 + B1x2 + B2x + B3,  where Bi  =  Bi0f0 +Bi7f7,   i=1,  2,  3.

Date received: August 2, 2001