Bicomplex number

In abstract algebra, a bicomplex number is a pair of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate formula_1, and the product of two bicomplex numbers as
Then the bicomplex norm is given by
The bicomplex numbers form a commutative algebra over C of dimension two, which is isomorphic to the direct sum of algebras .
The product of two bicomplex numbers yields a quadratic form value that is the product of the individual quadratic forms of the numbers: 
a verification of this property of the quadratic form of a product refers to the Brahmagupta–Fibonacci identity. This property of the quadratic form of a bicomplex number indicates that these numbers form a composition algebra. In fact, bicomplex numbers arise at the binarion level of the Cayley–Dickson construction based on ℂ with norm z2.
The general bicomplex number can be represented by the matrix formula_4, which has determinant formula_5. Thus, the composing property of the quadratic form concurs with the composing property of the determinant.
As a real algebra.
Bicomplex numbers form an algebra over C of dimension two, and since C is of dimension two over R, the bicomplex numbers are an algebra over R of dimension four. In fact the real algebra is older than the complex one; it was labelled "tessarines" in 1848 while the complex algebra was not introduced until 1892.
A basis for the tessarine 4-algebra over R specifies "z" = 1 and "z" = −"i", giving the matrices 
formula_6, which multiply according to the table given. When the identity matrix is identified with 1, then a tessarine "t" = "w" + "z j" .
History.
The subject of multiple imaginary units was examined in the 1840s. In a long series "On quaternions, or on a new system of imaginaries in algebra" beginning in 1844 in Philosophical Magazine, William Rowan Hamilton communicated a system multiplying according to the quaternion group. In 1848 Thomas Kirkman reported on his correspondence with Arthur Cayley regarding equations on the units determining a system of hypercomplex numbers.
Tessarines.
In 1848 James Cockle introduced the tessarines in a series of articles in "Philosophical Magazine".
A tessarine is a hypercomplex number of the form
where formula_8
Cockle used tessarines to isolate the hyperbolic cosine series and the hyperbolic sine series in the exponential series. He also showed how zero divisors arise in tessarines, inspiring him to use the term "impossibles". The tessarines are now best known for their subalgebra of real tessarines formula_9, 
also called split-complex numbers, which express the parametrization of the unit hyperbola.
Bicomplex numbers.
In a 1892 "Mathematische Annalen" paper, Corrado Segre introduced bicomplex numbers, which form an algebra isomorphic to the tessarines.
Segre read W. R. Hamilton's "Lectures on Quaternions" (1853) and the works of W. K. Clifford. Segre used some of Hamilton's notation to develop his system of bicomplex numbers: Let "h" and "i" be elements that square to −1 and that commute. Then, presuming associativity of multiplication, the product "hi" must square to +1. The algebra constructed on the basis is then the same as James Cockle's tessarines, represented using a different basis. Segre noted that elements
When bicomplex numbers are expressed in terms of the basis , their equivalence with tessarines is apparent. Looking at the linear representation of these isomorphic algebras shows agreement in the fourth dimension when the negative sign is used; consider the sample product given above under linear representation.
Quotient rings of polynomials.
One comparison of bicomplex numbers and tessarines uses the polynomial ring R["X","Y"], where . The ideal formula_11 then provides a quotient ring representing tessarines. In this quotient ring approach, elements of the tessarines correspond to cosets with respect to the ideal "A". Similarly, the ideal formula_12 produces a quotient representing bicomplex numbers.
A generalization of this approach uses the free algebra in two "non-commuting" indeterminates "X" and "Y". Consider these three second degree polynomials formula_13. Let "A" be the ideal generated by them. Then the quotient ring is isomorphic to the ring of tessarines.
To see that formula_14  note that
Now consider the alternative ideal "B" generated by formula_18.
In this case one can prove formula_19. The ring isomorphism involves a change of basis exchanging formula_20.
Alternatively, suppose the field C of ordinary complex numbers is presumed given, and C["X"] is the ring of polynomials in "X" with complex coefficients. Then the quotient is another presentation of bicomplex numbers.
Polynomial roots.
Write and represent elements of it by ordered pairs ("u","v") of complex numbers. Since the algebra of tessarines T is isomorphic to 2C, the rings of polynomials T[X] and 2C["X"] are also isomorphic, however polynomials in the latter algebra split:
In consequence, when a polynomial equation formula_22 in this algebra is set, it reduces to two polynomial equations on C. If the degree is "n", then there are "n" roots for each equation: formula_23
Any ordered pair formula_24 from this set of roots will satisfy the original equation in 2C["X"], so it has "n"2 roots.
Due to the isomorphism with T["X"], there is a correspondence of polynomials and a correspondence of their roots. Hence the tessarine polynomials of degree "n" also have "n"2 roots, counting multiplicity of roots.
Applications.
Tessarines have been applied in digital signal processing.