Gaussian integer

Template:Short description Template:Distinguish In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as <math>\mathbf{Z}[i]</math> or <math>\Z[i].</math><ref name="Fraleigh 1976 286">Template:Harvtxt</ref>

Gaussian integers share many properties with integers: they form a Euclidean domain, and thus have a Euclidean division and a Euclidean algorithm; this implies unique factorization and many related properties. However, Gaussian integers do not have a total order that respects arithmetic.

Gaussian integers are algebraic integers and form the simplest ring of quadratic integers.

Gaussian integers are named after the German mathematician Carl Friedrich Gauss.

File:Gaussian integer lattice.svg

Gaussian integers as lattice points in the complex plane

Basic definitions

The Gaussian integers are the set<ref name="Fraleigh 1976 286"/>

<math>\mathbf{Z}[i]=\{a+bi \mid a,b\in \mathbf{Z} \}, \qquad \text{ where } i^2 = -1.</math>

In other words, a Gaussian integer is a complex number such that its real and imaginary parts are both integers. Since the Gaussian integers are closed under addition and multiplication, they form a commutative ring, which is a subring of the field of complex numbers. It is thus an integral domain.

When considered within the complex plane, the Gaussian integers constitute the Template:Math-dimensional integer lattice.

The conjugate of a Gaussian integer Template:Math is the Gaussian integer Template:Math.

The norm of a Gaussian integer is its product with its conjugate.

<math>N(a+bi) = (a+bi)(a-bi) = a^2+b^2.</math>

The norm of a Gaussian integer is thus the square of its absolute value as a complex number. The norm of a Gaussian integer is a nonnegative integer, which is a sum of two squares. By the sum of two squares theorem, a norm cannot have a factor <math>p^k</math> in its prime decomposition where <math>p \equiv 3 \pmod 4</math> and <math>k</math> is odd (in particular, a norm is not itself congruent to 3 modulo 4).

The norm is multiplicative, that is, one has<ref>Template:Harvtxt</ref>

<math>N(zw) = N(z)N(w),</math>

for every pair of Gaussian integers Template:Math. This can be shown directly, or by using the multiplicative property of the modulus of complex numbers.

The units of the ring of Gaussian integers (that is the Gaussian integers whose multiplicative inverse is also a Gaussian integer) are precisely the Gaussian integers with norm 1, that is, 1, −1, Template:Math and Template:Math.<ref>Template:Harvtxt</ref>

Euclidean division

File:Gauss-euklid.svg

Visualization of maximal distance to some Gaussian integer

Gaussian integers have a Euclidean division (division with remainder) similar to that of integers and polynomials. This makes the Gaussian integers a Euclidean domain, and implies that Gaussian integers share with integers and polynomials many important properties such as the existence of a Euclidean algorithm for computing greatest common divisors, Bézout's identity, the principal ideal property, Euclid's lemma, the unique factorization theorem, and the Chinese remainder theorem, all of which can be proved using only Euclidean division.

A Euclidean division algorithm takes, in the ring of Gaussian integers, a dividend Template:Math and divisor Template:Math, and produces a quotient Template:Math and remainder Template:Math such that

<math>a=bq+r\quad \text{and} \quad N(r)<N(b).</math>

In fact, one may make the remainder smaller:

<math>a=bq+r\quad \text{and} \quad N(r)\le \frac{N(b)}{2}.</math>

Template:AnchorEven with this better inequality, the quotient and the remainder are not necessarily unique, but one may refine the choice to ensure uniqueness.

To prove this, one may consider the complex number quotient Template:Math. There are unique integers Template:Math and Template:Math such that Template:Math and Template:Math, and thus Template:Math. Taking Template:Math, one has

<math>a = bq + r,</math>

with

<math>r=b\bigl(x-m+ i(y-n)\bigr), </math>

and

<math>N(r)\le \frac{N(b)}{2}.</math>

The choice of Template:Math and Template:Math in a semi-open interval is required for uniqueness. This definition of Euclidean division may be interpreted geometrically in the complex plane (see the figure), by remarking that the distance from a complex number Template:Mvar to the closest Gaussian integer is at most Template:Math.<ref>Template:Harvtxt</ref>

Principal ideals

Since the ring Template:Math of Gaussian integers is a Euclidean domain, Template:Math is a principal ideal domain, which means that every ideal of Template:Mvar is principal. Explicitly, an ideal Template:Mvar is a subset of a ring Template:Mvar such that every sum of elements of Template:Mvar and every product of an element of Template:Mvar by an element of Template:Mvar belong to Template:Mvar. An ideal is principal if it consists of all multiples of a single element Template:Math, that is, it has the form

<math>\{gx\mid x\in G\}.</math>

In this case, one says that the ideal is generated by Template:Math or that Template:Math is a generator of the ideal.

Every ideal Template:Math in the ring of the Gaussian integers is principal, because, if one chooses in Template:Math a nonzero element Template:Math of minimal norm, for every element Template:Math of Template:Math, the remainder of Euclidean division of Template:Math by Template:Math belongs also to Template:Math and has a norm that is smaller than that of Template:Math; because of the choice of Template:Math, this norm is zero, and thus the remainder is also zero. That is, one has Template:Math, where Template:Math is the quotient.

For any Template:Math, the ideal generated by Template:Math is also generated by any associate of Template:Math, that is, Template:Math; no other element generates the same ideal. As all the generators of an ideal have the same norm, the norm of an ideal is the norm of any of its generators.

Template:AnchorIn some circumstances, it is useful to choose, once for all, a generator for each ideal. There are two classical ways for doing that, both considering first the ideals of odd norm. If the Template:Math has an odd norm Template:Math, then one of Template:Math and Template:Math is odd, and the other is even. Thus Template:Math has exactly one associate with a real part Template:Math that is odd and positive. In his original paper, Gauss made another choice, by choosing the unique associate such that the remainder of its division by Template:Math is one. In fact, as Template:Math, the norm of the remainder is not greater than 4. As this norm is odd, and 3 is not the norm of a Gaussian integer, the norm of the remainder is one, that is, the remainder is a unit. Multiplying Template:Math by the inverse of this unit, one finds an associate that has one as a remainder, when divided by Template:Math.

If the norm of Template:Math is even, then either Template:Math or Template:Math, where Template:Math is a positive integer, and Template:Math is odd. Thus, one chooses the associate of Template:Math for getting a Template:Math which fits the choice of the associates for elements of odd norm.

Gaussian primes

As the Gaussian integers form a principal ideal domain, they also form a unique factorization domain. This implies that a Gaussian integer is irreducible (that is, it is not the product of two non-units) if and only if it is prime (that is, it generates a prime ideal).

The prime elements of Template:Math are also known as Gaussian primes. An associate of a Gaussian prime is also a Gaussian prime. The conjugate of a Gaussian prime is also a Gaussian prime (this implies that Gaussian primes are symmetric about the real and imaginary axes).

A positive integer is a Gaussian prime if and only if it is a prime number that is congruent to 3 modulo 4 (that is, it may be written Template:Math, with Template:Math a nonnegative integer) (sequence A002145 in the OEIS). The other prime numbers are not Gaussian primes, but each is the product of two conjugate Gaussian primes.

A Gaussian integer Template:Math is a Gaussian prime if and only if either:

• one of Template:Math is zero and the absolute value of the other is a prime number of the form Template:Math (with Template:Mvar a nonnegative integer), or
• both are nonzero and Template:Math is a prime number (which will never be of the form Template:Math).

In other words, a Gaussian integer Template:Math is a Gaussian prime if and only if either its norm is a prime number, or Template:Math is the product of a unit (Template:Math) and a prime number of the form Template:Math.

It follows that there are three cases for the factorization of a prime natural number Template:Math in the Gaussian integers:

• If Template:Math is congruent to 3 modulo 4, then it is a Gaussian prime; in the language of algebraic number theory, Template:Math is said to be inert in the Gaussian integers.
• If Template:Math is congruent to 1 modulo 4, then it is the product of a Gaussian prime by its conjugate, both of which are non-associated Gaussian primes (neither is the product of the other by a unit); Template:Math is said to be a decomposed prime in the Gaussian integers. For example, Template:Math and Template:Math.
• If Template:Math, we have Template:Math; that is, 2 is the product of the square of a Gaussian prime by a unit; it is the unique ramified prime in the Gaussian integers.

Unique factorization

As for every unique factorization domain, every Gaussian integer may be factored as a product of a unit and Gaussian primes, and this factorization is unique up to the order of the factors, and the replacement of any prime by any of its associates (together with a corresponding change of the unit factor).

If one chooses, once for all, a fixed Gaussian prime for each equivalence class of associated primes, and if one takes only these selected primes in the factorization, then one obtains a prime factorization which is unique up to the order of the factors. With the choices described above, the resulting unique factorization has the form

<math>u(1+i)^{e_0}{p_1}^{e_1}\cdots {p_k}^{e_k},</math>

where Template:Math is a unit (that is, Template:Math), Template:Math and Template:Math are nonnegative integers, Template:Math are positive integers, and Template:Math are distinct Gaussian primes such that, depending on the choice of selected associates,

• either Template:Math with Template:Math odd and positive, and Template:Math even,
• or the remainder of the Euclidean division of Template:Math by Template:Math equals 1 (this is Gauss's original choice<ref>Template:Harvtxt</ref>).

An advantage of the second choice is that the selected associates behave well under products for Gaussian integers of odd norm. On the other hand, the selected associate for the real Gaussian primes are negative integers. For example, the factorization of 231 in the integers, and with the first choice of associates is Template:Nowrap, while it is Template:Nowrap with the second choice.

Gaussian rationals

The field of Gaussian rationals is the field of fractions of the ring of Gaussian integers. It consists of the complex numbers whose real and imaginary part are both rational.

The ring of Gaussian integers is the integral closure of the integers in the Gaussian rationals.

This implies that Gaussian integers are quadratic integers and that a Gaussian rational is a Gaussian integer, if and only if it is a solution of an equation

<math>x^2 +cx+d=0,</math>

with Template:Math and Template:Math integers. In fact Template:Math is solution of the equation

<math>x^2-2ax+a^2+b^2,</math>

and this equation has integer coefficients if and only if Template:Math and Template:Math are both integers.

Template:AnchorGreatest common divisor

As for any unique factorization domain, a greatest common divisor (gcd) of two Gaussian integers Template:Math is a Gaussian integer Template:Math that is a common divisor of Template:Math and Template:Math, which has all common divisors of Template:Math and Template:Math as divisor. That is (where Template:Math denotes the divisibility relation),

• Template:Math and Template:Math, and
• Template:Math and Template:Math implies Template:Math.

Thus, greatest is meant relatively to the divisibility relation, and not for an ordering of the ring (for integers, both meanings of greatest coincide).

More technically, a greatest common divisor of Template:Math and Template:Math is a generator of the ideal generated by Template:Math and Template:Math (this characterization is valid for principal ideal domains, but not, in general, for unique factorization domains).

The greatest common divisor of two Gaussian integers is not unique, but is defined up to the multiplication by a unit. That is, given a greatest common divisor Template:Math of Template:Math and Template:Math, the greatest common divisors of Template:Math and Template:Math are Template:Math, and Template:Math.

There are several ways for computing a greatest common divisor of two Gaussian integers Template:Math and Template:Math. When one knows the prime factorizations of Template:Math and Template:Math,

<math>a = i^k\prod_m {p_m}^{\nu_m}, \quad b = i^n\prod_m {p_m}^{\mu_m},</math>

where the primes Template:Math are pairwise non associated, and the exponents Template:Math non-associated, a greatest common divisor is

<math>\prod_m {p_m}^{\lambda_m},</math>

with

<math>\lambda_m = \min(\nu_m, \mu_m).</math>

Unfortunately, except in simple cases, the prime factorization is difficult to compute, and Euclidean algorithm leads to a much easier (and faster) computation. This algorithm consists of replacing of the input Template:Math by Template:Math, where Template:Math is the remainder of the Euclidean division of Template:Math by Template:Math, and repeating this operation until getting a zero remainder, that is a pair Template:Math. This process terminates, because, at each step, the norm of the second Gaussian integer decreases. The resulting Template:Math is a greatest common divisor, because (at each step) Template:Math and Template:Math have the same divisors as Template:Math and Template:Math, and thus the same greatest common divisor.

This method of computation works always, but is not as simple as for integers because Euclidean division is more complicated. Therefore, a third method is often preferred for hand-written computations. It consists in remarking that the norm Template:Math of the greatest common divisor of Template:Math and Template:Math is a common divisor of Template:Math, Template:Math, and Template:Math. When the greatest common divisor Template:Math of these three integers has few factors, then it is easy to test, for common divisor, all Gaussian integers with a norm dividing Template:Math.

For example, if Template:Math, and Template:Math, one has Template:Math, Template:Math, and Template:Math. As the greatest common divisor of the three norms is 2, the greatest common divisor of Template:Math and Template:Math has 1 or 2 as a norm. As a gaussian integer of norm 2 is necessary associated to Template:Math, and as Template:Math divides Template:Math and Template:Math, then the greatest common divisor is Template:Math.

If Template:Math is replaced by its conjugate Template:Math, then the greatest common divisor of the three norms is 34, the norm of Template:Math, thus one may guess that the greatest common divisor is Template:Math, that is, that Template:Math. In fact, one has Template:Math.

Template:AnchorCongruences and residue classes

Given a Gaussian integer Template:Math, called a modulus, two Gaussian integers Template:Math are congruent modulo Template:Math, if their difference is a multiple of Template:Math, that is if there exists a Gaussian integer Template:Math such that Template:Math. In other words, two Gaussian integers are congruent modulo Template:Math, if their difference belongs to the ideal generated by Template:Math. This is denoted as Template:Math.

The congruence modulo Template:Math is an equivalence relation (also called a congruence relation), which defines a partition of the Gaussian integers into equivalence classes, called here congruence classes or residue classes. The set of the residue classes is usually denoted Template:Math, or Template:Math, or simply Template:Math.

The residue class of a Gaussian integer Template:Math is the set

<math> \bar a := \left\{ z \in \mathbf{Z}[i] \mid z \equiv a \pmod{z_0} \right\}</math>

of all Gaussian integers that are congruent to Template:Math. It follows that Template:Math if and only if Template:Math.

Addition and multiplication are compatible with congruences. This means that Template:Math and Template:Math imply Template:Math and Template:Math. This defines well-defined operations (that is independent of the choice of representatives) on the residue classes:

<math>\bar a + \bar b := \overline{a+b}\quad \text{and}\quad \bar a \cdot\bar b := \overline{ab}.</math>

With these operations, the residue classes form a commutative ring, the quotient ring of the Gaussian integers by the ideal generated by Template:Math, which is also traditionally called the residue class ring modulo Template:Math (for more details, see Quotient ring).

Examples

• There are exactly two residue classes for the modulus Template:Math, namely Template:Math (all multiples of Template:Math), and Template:Math, which form a checkerboard pattern in the complex plane. These two classes form thus a ring with two elements, which is, in fact, a field, the unique (up to an isomorphism) field with two elements, and may thus be identified with the integers modulo 2. These two classes may be considered as a generalization of the partition of integers into even and odd integers. Thus one may speak of even and odd Gaussian integers (Gauss divided further even Gaussian integers into even, that is divisible by 2, and half-even).
• For the modulus 2 there are four residue classes, namely Template:Math. These form a ring with four elements, in which Template:Math for every Template:Math. Thus this ring is not isomorphic with the ring of integers modulo 4, another ring with four elements. One has Template:Math, and thus this ring is not the finite field with four elements, nor the direct product of two copies of the ring of integers modulo 2.
• For the modulus Template:Math there are eight residue classes, namely Template:Math, whereof four contain only even Gaussian integers and four contain only odd Gaussian integers.

Describing residue classes

File:Gauss-Restklassen-wiki.png

All 13 residue classes with their minimal residues (blue dots) in the square Template:Math (light green background) for the modulus Template:Math. One residue class with Template:Math is highlighted with yellow/orange dots.

Given a modulus Template:Math, all elements of a residue class have the same remainder for the Euclidean division by Template:Math, provided one uses the division with unique quotient and remainder, which is described above. Thus enumerating the residue classes is equivalent with enumerating the possible remainders. This can be done geometrically in the following way.

In the complex plane, one may consider a square grid, whose squares are delimited by the two lines

<math>\begin{align}

V_s&=\left\{ \left. z_0\left(s-\tfrac12 +ix\right) \right\vert x\in \mathbf R\right\} \quad \text{and} \\ H_t&=\left\{ \left. z_0\left(x+i\left(t-\tfrac12\right)\right) \right\vert x\in \mathbf R\right\}, \end{align}</math>

with Template:Math and Template:Math integers (blue lines in the figure). These divide the plane in semi-open squares (where Template:Math and Template:Math are integers)

<math>Q_{mn}=\left\{(s+it)z_0 \left\vert s \in \left [m - \tfrac12, m + \tfrac12\right), t \in \left[n - \tfrac12, n + \tfrac12 \right)\right.\right\}.</math>

The semi-open intervals that occur in the definition of Template:Math have been chosen in order that every complex number belong to exactly one square; that is, the squares Template:Math form a partition of the complex plane. One has

<math>Q_{mn} = (m+in)z_0+Q_{00}=\left\{(m+in)z_0+z\mid z\in Q_{00}\right\}.</math>

This implies that every Gaussian integer is congruent modulo Template:Math to a unique Gaussian integer in Template:Math (the green square in the figure), which its remainder for the division by Template:Math. In other words, every residue class contains exactly one element in Template:Math.

The Gaussian integers in Template:Math (or in its boundary) are sometimes called minimal residues because their norm are not greater than the norm of any other Gaussian integer in the same residue class (Gauss called them absolutely smallest residues).

From this one can deduce by geometrical considerations, that the number of residue classes modulo a Gaussian integer Template:Math equals its norm Template:Math (see below for a proof; similarly, for integers, the number of residue classes modulo Template:Math is its absolute value Template:Math).

Residue class fields

The residue class ring modulo a Gaussian integer Template:Math is a field if and only if <math>z_0</math> is a Gaussian prime.

If Template:Math is a decomposed prime or the ramified prime Template:Math (that is, if its norm Template:Math is a prime number, which is either 2 or a prime congruent to 1 modulo 4), then the residue class field has a prime number of elements (that is, Template:Math). It is thus isomorphic to the field of the integers modulo Template:Math.

If, on the other hand, Template:Math is an inert prime (that is, Template:Math is the square of a prime number, which is congruent to 3 modulo 4), then the residue class field has Template:Math elements, and it is an extension of degree 2 (unique, up to an isomorphism) of the prime field with Template:Math elements (the integers modulo Template:Math).

Primitive residue class group and Euler's totient function

Many theorems (and their proofs) for moduli of integers can be directly transferred to moduli of Gaussian integers, if one replaces the absolute value of the modulus by the norm. This holds especially for the primitive residue class group (also called [[multiplicative group of integers modulo n|multiplicative group of integers modulo Template:Math]]) and Euler's totient function. The primitive residue class group of a modulus Template:Math is defined as the subset of its residue classes, which contains all residue classes Template:Math that are coprime to Template:Math, i.e. Template:Math. Obviously, this system builds a multiplicative group. The number of its elements shall be denoted by Template:Math (analogously to Euler's totient function Template:Math for integers Template:Math).

For Gaussian primes it immediately follows that Template:Math and for arbitrary composite Gaussian integers

<math>z = i^k\prod_m {p_m}^{\nu_m}</math>

Euler's product formula can be derived as

<math>\phi(z) =\prod_{m\, (\nu_m > 0)} \bigl|{p_m}^{\nu_m}\bigr|^2 \left( 1 - \frac 1{|p_m|{}^2} \right) = |z|^2\prod_{p_m|z}\left( 1 - \frac 1{|p_m|{}^2} \right)</math>

where the product is to build over all prime divisors Template:Math of Template:Math (with Template:Math). Also the important theorem of Euler can be directly transferred:

For all Template:Math with Template:Math, it holds that Template:Math.

Historical background

The ring of Gaussian integers was introduced by Carl Friedrich Gauss in his second monograph on quartic reciprocity (1832).<ref>Template:Harvtxt</ref> The theorem of quadratic reciprocity (which he had first succeeded in proving in 1796) relates the solvability of the congruence Template:Math to that of Template:Math. Similarly, cubic reciprocity relates the solvability of Template:Math to that of Template:Math, and biquadratic (or quartic) reciprocity is a relation between Template:Math and Template:Math. 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

File:Gauss-primes-768x768.png

The distribution of the small Gaussian primes in the complex plane

Most of the unsolved problems are related to distribution of Gaussian primes in the plane.

• 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 Template:Math?<ref>Ribenboim, Ch.III.4.D Ch. 6.II, Ch. 6.IV (Hardy & Littlewood's conjecture E and F)</ref>
• Is it possible to walk to infinity using the Gaussian primes as stepping stones and taking steps of a uniformly bounded length? This is known as the Gaussian moat problem; it was posed in 1962 by Basil Gordon and remains unsolved.<ref>Template:Cite journal</ref><ref>Template:Cite book</ref>

See also

• Algebraic integer
• Cyclotomic field
• Eisenstein integer
• Eisenstein prime
• Hurwitz quaternion
• Proofs of Fermat's theorem on sums of two squares
• Proofs of quadratic reciprocity
• Quadratic integer
• Splitting of prime ideals in Galois extensions describes the structure of prime ideals in the Gaussian integers
• Table of Gaussian integer factorizations

References

• Template:Citation; reprinted in Werke, Georg Olms Verlag, Hildesheim, 1973, pp. 93–148. A German translation of this paper is available online in ″H. Maser (ed.): Carl Friedrich Gauss’ Arithmetische Untersuchungen über höhere Arithmetik. Springer, Berlin 1889, pp. 534″.
• Template:Citation
• Template:Cite journal
• Template:Cite book
• Template:Cite journal

External links

• IMO Compendium text on quadratic extensions and Gaussian Integers in problem solving
• Keith Conrad, The Gaussian Integers.