Gaussian integer

In number theory, 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]. The Gaussian integers are a special case of the quadratic integers. This domain does not have a total ordering that respects arithmetic.

Gaussian integers as lattice points in the complex plane

Formally, Gaussian integers are the set

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

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

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

(Where the overline over "a+bi" refers to the complex conjugate.)

The norm is multiplicative, i.e.

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

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. If x is a Gaussian integer, the four numbers x, ix, −x, and −ix are called the associates of x.

The prime elements of Z[i] are also known as Gaussian primes. An associate of a Gaussian prime is also a Gaussian prime.

The positive integer Gaussian primes are (sequence A002145 in the OEIS). (It is a common error to refer to only these positive integers as "the Gaussian primes" when in fact this term refers to all the Gaussian primes)

Some of the 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 $-(4n+3)$ (where $n\geq 0$)
• or both are nonzero and $a^{2}+b^{2}$ is prime.

The following elaborates on these conditions.

2 is a special case (in the language of algebraic number theory, 2 is the only ramified prime in Z[i]).

The integer 2 factors as $2=i(1-i)^{2}$ when considered as a Gaussian integer. It is the only prime integer divisible by the square of a Gaussian prime.

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 $g|g{\bar {g}}=N(g)$. 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 $N(g)=g{\bar {g}}|p_{i}^{2}$. 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 ${\overline {g}}=p{\overline {u}}$ 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 his second monograph on quartic reciprocity (1832) (see [2]). The theorem of quadratic reciprocity (which he had first succeeded in proving in 1796) relates the solvability of the congruence x² ≡ q (mod p) to that of x² ≡ p (mod q). Similarly, cubic reciprocity relates the solvability of x³ ≡ q (mod p) to that of x³ ≡ p (mod q), and biquadratic (or quartic) reciprocity is a relation between x⁴ ≡ q (mod p) and x⁴ ≡ p (mod q). Gauss discovered that the law of biquadratic reciprocity and its supplements were more easily stated and proved as statements about "whole complex numbers" (i.e. the Gaussian integers) than they are as statements about ordinary whole numbers (i.e. the integers).

In a footnote he notes that the Eisenstein integers are the natural domain for stating and proving results on cubic reciprocity and indicates that similar extensions of the integers are the appropriate domains for studying higher reciprocity laws.

This paper not only introduced the Gaussian integers and proved they are a unique factorization domain, it also introduced the terms norm, unit, primary, and associate, which are now standard in algebraic number theory.

Unsolved problems

Gauss's circle problem does not deal with the Gaussian integers per se, but instead asks for the number of lattice points inside a circle of a given radius centered at the origin. This is equivalent to determining the number of Gaussian integers with norm less than a given value.

There are also conjectures and unsolved problems about the Gaussian primes. Two of them are:

The real and imaginary axes have the infinite set of Gaussian primes 3, 7, 11, 19, ... and their associates. Are there any other lines that have infinitely many Gaussian primes on them? In particular, are there infinitely many Gaussian primes of the form 1+ki?

Is it possible to walk to infinity using the Gaussian primes as stepping stones and taking steps of bounded length?

See also

• Hurwitz quaternion
• 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
• Quadratic integer
• Table of Gaussian integer factorizations

References

• 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)
• Ribenboim, Paulo (1996). "The New Book of Prime Number Records" (Document). New York: Springer. {{cite document}}: Unknown parameter |isbn= ignored (help)CS1 maint: postscript (link)

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.
• [3] Complex Gaussian Integers for 'Gaussian Graphics'
• IMO Compendium text on quadratic extensions and Gaussian Integers in problem solving
• Weisstein, Eric W. "Moat-Crossing Problem". MathWorld.
• Gethner, Ellen; Wagon, Stan; Wick, Brian (1998). "A Stroll Through the Gaussian Primes" (PDF). American Mathematical Monthly. 105 (4): 327–337. {{cite journal}}: Unknown parameter |month= ignored (help)CS1 maint: postscript (link)
• Weisstein, Eric W. "Landau's Problems". MathWorld.