integer factorization in Z(sqrt(d)) where Z(sqrt(d)) is a Euclidean ring
integer, list or set of integers in Z⁡d
integer where d is a Euclidean ring
Important: The numtheory package has been deprecated. Use the superseding command NumberTheory[FactorNormEuclidean] instead.
The factorEQ function returns the integer factorization of m in the Euclidean ring Z⁡d.
Given integers a and b of Z⁡d, with b≠0, there is an integer q such that a=q⁢b+r, norm⁡r<norm⁡b is true in Z⁡d. In these circumstances we say that there is a Euclidean algorithm in Z⁡d and that the ring is Euclidean.
Euclidean quadratic number fields have been completely determined. They are Z⁡d where d = -1, -2, -3, -7, -11, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, and 73.
When d=2,3mod4, all integers of Z⁡d have the form a+b⁢d, where a and b are rational integers. When d=1mod4, all integers of Z⁡d are of the form a+b⁢d2 where a and b are rational integers and of the same parity.
The answer is in the form: ±1⁢u⁢f1e1⁢...⁢fnen such that m=±1⋅⁢u⁢f1e1⁢…⁢⁢fnen where f1,…,fn are distinct prime factors of m, e1,…,en are non-negative integer numbers, u is a unit in Z⁡d. For real Euclidean quadratic rings, i.e. d > 0, u is represented under the form wn or w&conjugate0;n or −wn or −w&conjugate0;n where w is the fundamental unit, and n is a positive integer.
The expand function may be applied to cause the factors to be multiplied together again.
Download Help Document
What kind of issue would you like to report? (Optional)