Gaussian integer

A Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. This domain cannot be turned into an ordered ring, since it contains a square root of −1.

Gaussian integers as lattice points in the complex plane

Formally, Gaussian integers are the set

$\{a+bi\mid a,b\in \mathbb {Z} \}.$

The norm of a Gaussian integer is the natural number defined as

N(a + bi) = a² + b².

The norm is multiplicative, i.e.

N(z·w) = N(z)·N(w).

The units of Z[i] are therefore precisely those elements with norm 1, i.e. the elements

1, −1, i and −i.

The prime elements of Z[i] are also known as Gaussian primes. Some prime numbers (which, by contrast, are sometimes referred to as "rational primes") are not Gaussian primes; for example 2 = (1 + i)(1 − i) and 5 = (2 + i)(2 − i). Those rational primes which are congruent to 3 (mod 4) are Gaussian primes; those which are congruent to 1 (mod 4) are not. This is because primes of the form 4k + 1 can always be written as the sum of two squares (Fermat's theorem), so we have

p = a² + b² = (a + bi)(a − bi).

If the norm of a Gaussian integer z is a prime number, then z must be a Gaussian prime, since every non-trivial factorization of z would yield a non-trivial factorization of the norm and irreducible numbers are prime, since Z[i] is a Euclidean domain. So for example 2 + 3i is a Gaussian prime since its norm is 4 + 9 = 13.

The ring of Gaussian integers is the integral closure of Z in the field of Gaussian rationals Q(i) consisting of the complex numbers whose real and imaginary part are both rational.

It is easy to see graphically that every complex number is within ${\frac {\sqrt {2}}{2}}$ units of a Gaussian integer. Put another way, every complex number (and hence every Gaussian integer) is within ${\frac {\sqrt {2}}{2}}N(z)$ units of some multiple of z, where z is any Gaussian integer; this turns Z(i) into a Euclidean domain, where v(z) = N(z).

Historical background

The ring of Gaussian integers was introduced by Carl Friedrich Gauss in 1829 - 1831 (see [1]) while studying reciprocity laws which are generalisations of the theorem of quadratic reciprocity which he had first succeeded in proving in 1796. In particular, he was looking for relationships between p and q such that q should be a cubic residue of p (i.e. $x^{3}\equiv q({\rm {mod}}\ p)$) or such that q should be a biquadratic residue of p (i.e. $x^{4}\equiv q({\rm {mod}}\ p)$). During this research he discovered that some results were more easily provable by working in the ring of Gaussian integers, rather than the ordinary integers.

He developed the properties of factorisation and proved the uniqueness of factorisation into primes in Z[i], and despite publishing little, he made some comments which indicate that he was aware of the significance of Eisenstein integers in stating and proving results on cubic reciprocity.

See also

• Eisenstein integer
• Proofs of Fermat's theorem on sums of two squares
• Proofs of quadratic reciprocity
• Splitting of prime ideals in Galois extensions describes the structure of prime ideals in the Gaussian integers

Bibliography

• C. F. Gauss, Theoria residuorum biquadraticorum. Commentatio secunda., Comm. Soc. Reg. Sci. Gottingen 7 (1832) 1-­34; reprinted in Werke, Georg Olms Verlag, Hildesheim, 1973, pp. 93-­148.
• From Numbers to Rings: The Early History of Ring Theory, by Israel Kleiner (Elem. Math. 53 (1998) 18 – 35)

External links

• http://www.alpertron.com.ar/GAUSSIAN.HTM is a Java applet that evaluates expressions containing Gaussian integers and factors them into Gaussian primes.
• http://www.alpertron.com.ar/GAUSSPR.HTM is a Java applet that features a graphical view of Gaussian primes.
• [2] Complex Gaussian Integers for 'Gaussian Graphics'
• IMO Compendium text on quadratic extensions and Gaussian Integers in problem solving