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Eisenstein Integer


The Eisenstein integers, sometimes also called the Eisenstein-Jacobi integers (Finch 2003, p. 601), are numbers of the form a+bomega, where a and b are normal integers,

 omega=1/2(-1+isqrt(3))
(1)

is one of the roots of z^3=1, the others being 1 and

 omega^2=1/2(-1-isqrt(3)).
(2)

The sums, differences, and products of Eisenstein integers is another Eisenstein integer.

Eisenstein integers are complex numbers that are members of the imaginary quadratic field Q(sqrt(-3)), which is precisely the ring Z[omega] (Wagon 1991, p. 320). The field of Eisenstein integers has the six units (or roots of unity), namely +/-1, +/-omega, and +/-omega^2 (Wagon 1991, p. 320; Guy 1994, p. 35).

EisensteinIntegers

Every nonzero Eisenstein integer has a unique (up to ordering) factorization up to associates, where associates are Eisenstein integers related to the given Eisenstein integer by rotations of multiples of 60 degrees in the complex plane. Specifically, any nonzero Eisenstein integer is uniquely the product of powers of -1, omega, and the "positive" Eisenstein primes, where the "positive" Eisenstein integers are those falling within the triangular wedge illustrated above (Conway and Guy 1996).

The analog of Fermat's theorem for Eisenstein integers is that a prime number p can be written in the form

 a^2-ab+b^2=(a+bomega)(a+bomega^2)
(3)

iff 3p+1. These are precisely the primes of the form 3m^2+n^2 (Conway and Guy 1996).

Every Eisenstein integer is within a distance |n|/sqrt(3) of some multiple of a given Eisenstein integer n.

Dörrie (1965) uses the alternative notation

J=1/2(1+isqrt(3))
(4)
O=1/2(1-isqrt(3)).
(5)

for -omega^2 and -omega, and calls numbers of the form aJ+bO G-numbers. O and J satisfy

J+O=1
(6)
JO=1
(7)
J^2+O=0
(8)
O^2+J=0
(9)
J^3=-1
(10)
O^3=-1.
(11)

See also

Eisenstein Prime, Eisenstein Unit, Gaussian Integer, Integer

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References

Bachmann, P. Allgemeine Arithmetik der Zahlkörper. p. 142.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, p. 88, 2003.Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 220-223, 1996.Cox, D. A. §4A in Primes of the Form x2+ny2: Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1989.Dörrie, H. "The Fermat-Gauss Impossibility Theorem." §21 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 96-104, 1965.Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.Guy, R. K. "Gaussian Primes. Eisenstein-Jacobi Primes." §A16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 33-36, 1994.Hardy, G. H. and Wright, E. M. "The Integers of k(rho)." §12.9 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 187-189, 1979.Riesel, H. Appendix 4 in Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, 1994.Wagon, S. "Eisenstein Primes." §9.8 in Mathematica in Action. New York: W. H. Freeman, pp. 319-323, 1991.

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Eisenstein Integer

Cite this as:

Weisstein, Eric W. "Eisenstein Integer." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EisensteinInteger.html

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