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Kleinian integer

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Source updated January 20, 2022

In mathematical cryptography, a Kleinian integer is a complex number of the form m+n{\frac {1+{\sqrt {-7}}}{2}}, with m and n rational integers. They are named after Felix Klein.

The Kleinian integers form a ring called the Kleinian ring, which is the ring of integers in the imaginary quadratic field \mathbb {Q} ({\sqrt {-7}}). This ring is a unique factorization domain.

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Eisenstein integer

In mathematics, the Eisenstein integers, occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form z = a + b ω , {\displaystyle z=a+b\omega ,} where a and b are integers and ω = − 1 + i 3 2 = e i 2 π / 3 {\displaystyle \omega ={\frac {-1+i{\sqrt {3}}}{2}}=e^{i2\pi /3}} is a primitive (hence non-real) cube root of unity. The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers, which form a square lattice in the complex plane.

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