Gaussian prime number . Within the positive integers there is an infinite subset of the known prime integers, which are characterized by having exactly two divisors: 1 and the same number. In the same way in the set Z [i] of all integer Gaussian numbers there are elements that have the same behavior as prime integers, (commonly known as prime numbers).
Summary
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- 1 Definition
- 1 Examples
- 2 Decomposition theorem
- 3 Associated numbers
- 4 Proposition
- 5 References
- 6 Bibliographic source
- 7 External links
Definition
A Gaussian integer p is called a Gaussian prime if, in any of its decompositions p = s • t, as a product of two Gaussian integers, one of the factors s or t is divisor by the unit, that is, of the four only: 1, -1 , i, -i. [one]
Or equivalently, prime Gaussian number p is a non-zero Gaussian integer whose norm is greater than 1 and such a number cannot be decomposed as a product of two Gaussian numbers whose norms are strictly less than those of the number p.
Examples
p = 4 + i whose norm is N (p) = 17
p 1 = 2 -3i, of norm N (p 1 ) = 13
p 2 = 5 + 2i, whose norm is (Np 2 ) = 29
In general, all Gaussians who have as a norm a prime integer of finished form will be Gaussians.
Decomposition theorem
Any integer Gaussian number ω ≠ 0 can be decomposed into a product of prime Gaussian numbers
ω = l • m ••• q
the factors are prime Gaussian numbers that may not be different. This decomposition is univocal: for any other decomposition will include the same factors, except the order. This theorem is essentially the same as a number theory theorem about the decomposition of a compound integer as a product of prime numbers that could appear even with certain integer powers.
Associated numbers
Two integer Gaussian numbers are called associates if they differ from each other by a factor equal to a unit divisor, that is, b, -b, ib, -ib are associated integer Gaussian numbers if b is any integer Gaussian number.
Proposition
Every Gaussian prime is a divisor of a rational prime. [2]
As an example, the Gaussian prime 4 + i is a divisor of the rational prime 1, since (4 + i) • 84-i) = 17.
