What is eisenstein integer?

Eisenstein integers are a special type of complex number that are closely related to the triangular lattice in the complex plane. They are numbers of the form a + bω, where a and b are integers, and ω is a cube root of unity, specifically ω = (-1 + i√3)/2.

Eisenstein integers have some interesting properties, such as being a unique factorization domain, meaning that every Eisenstein integer can be expressed as a product of other Eisenstein integers in a unique way. They also form a ring, meaning that the sum, difference, and product of two Eisenstein integers is again an Eisenstein integer.

They have applications in number theory, particularly in studying prime numbers that can be expressed in the form a^2 + 3b^2, which is related to a conic curve in the complex plane.

Overall, Eisenstein integers are a fascinating area of study in mathematics, with connections to geometry, algebra, and number theory.