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 does not have a total ordering that respects arithmetic, since it contains imaginary numbers.

Formally, Gaussian integers are the set

$mathbb\{Z\}[i]=\{a+bi\; mid\; a,bin\; mathbb\{Z\}\; \}.$

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

$N\; left(a+bi\; right)\; =\; a^2+b^2\; =\; (a+bi)overline\{(a+bi)\}$

The norm is multiplicative, i.e.

$N(zcdot\; w)\; =\; N(z)cdot\; N(w)$.

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

1, −1, i and −i.

As a unique factorization domain

The Gaussian integers form a unique factorization domain with units 1, -1, i, and -i.

The prime elements of Z[i] are also known as Gaussian primes.

A Gaussian integer $a+bi$ is prime if and only if:

• one of a,b is zero and the other is a prime of the form $4n+3$ or its negative $-(4n+3)$
• or both are nonzero and $a^2+b^2$ is prime.

The following elaborates on these conditions.

The necessary conditions can be stated as following: a Gaussian integer is prime only when its norm is prime, or its norm is a square of a prime. This is because for any Gaussian integer $g$, notice $g\; |\; gbar\{g\}\; =N(g)$. Now $N(g)$ is an integer, and so can be factored as a product $p\_\{1\}p\_\{2\}cdots\; p\_\{n\}$ of rational primes, that is, as prime numbers in $mathbb\{Z\}$ by the fundamental theorem of arithmetic. By definition of prime, if $g$ is prime then it divides $p\_i$ for some $i$. Also, $bar\; g$ divides $overline\{p\_i\}=p\_i$, so $N(g)\; =\; gbar\{g\}\; |\; p\_\{i\}^\{2\}$. This gives only two options: either the norm of $g$ is prime, or the square of a prime.

If in fact $N(g)=p^2$ for some rational prime $p$, then both $g$ and $overline\{g\}$ divide $p^2$. Neither can be a unit, and so $g=pu$ and $overline\{g\}=poverline\{u\}$ where $u$ is a unit. This is to say that either $a=0$ or $b=0$, where $g=a+bi$

However, not every rational prime $p$ is a Gaussian prime. 2 is not because $2=(1+i)(1-i)$. Neither are primes of the form $4n+1$ because Fermat's theorem on sums of two squares assures us they can be written $a^2+b^2$ for integers $a$ and $b$, and $a^2+b^2\; =\; (a+bi)(a-bi)$. The only type of primes remaining are of the form $4n+3$.

Rational primes of the form $4n+3$ are also Gaussian primes. For suppose $g=p+0i$ for $p=4n+3$ a prime, and it can be factored $g=hk$. Then $p^2=N(g)=N(h)N(k)$. If the factorization is non-trivial, then $N(h)=N(k)=p$. But no sum of squares -- prime sum or not -- can be written $4n+3$. So the factorization must have been trivial and $g$ is a Gaussian prime.

Likewise $i$ times a rational prime of the form $4n+3$ is a Gaussian prime, but $i$ times a prime of the form $4n+1$ is not.

If $g$ is a Gaussian integer with prime norm, then $g$ is a Gaussian prime. This is because if $g=hk$, then $N(g)=N(h)N(k)$ and being prime one of $N(h)$, or $N(k)$ must be 1, hence one of $h$,$k$ must be a unit.

As an integral closure

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.

As a Euclidean domain

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) has a maximal distance of $frac\{sqrt\; 2\}\{2\}\; sqrt\{N(z)\}$ units to 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 ) 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. x³≡ q mod p) or such that q should be a biquadratic residue of p (i.e. x⁴≡ q 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

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