“Eisenstein Integer”的概念、定义、习题解题方法及翻译-数学辞典

单词

Eisenstein Integer

释义

Eisenstein Integer

The numbers , where

is one of the Roots of

, the others being 1 and

Eisenstein integers are members of the Quadratic Field

, and the Complex Numbers

. Every Eisenstein integer has a unique factorization. Specifically, any NonzeroEisenstein integer is uniquely the product of Powers of

, and the ``positive''Eisenstein Primes (Conway and Guy 1996). Every Eisenstein integer is within a distance

of some multiple of a given Eisenstein integer

Dörrie (1965) uses the alternative notation

			(1)
			(2)

for

and

, and calls numbers of the form

G-Number.

and

satisfy

			(3)
			(4)
			(5)
			(6)
			(7)
			(8)

The sum, difference, and products of

numbers are also

numbers. The norm of a

number is

(9)

The analog of Fermat's Theorem for Eisenstein integers is that a Prime Number can be written in theform

Iff

. These are precisely the Primes of the form

(Conway and Guy 1996).

See also Eisenstein Prime, Eisenstein Unit, Gaussian Integer, Integer

References

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 : 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.

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.

Riesel, H. Appendix 4 in Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, 1994.

Wagon, S. ``Eisenstein Primes.'' Mathematica in Action. New York: W. H. Freeman, pp. 278-279, 1991.

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