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
Z[i]
\Z[i].
Gaussian integers share many properties with integers: they form a Euclidean domain, and have thus a Euclidean division and a Euclidean algorithm; this implies unique factorization and many related properties. However, Gaussian integers do not have a total ordering 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.
The Gaussian integers are the set
Z[i]=\{a+bi\mida,b\inZ\}, wherei2=-1.
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 -dimensional integer lattice.
The conjugate of a Gaussian integer is the Gaussian integer .
The norm of a Gaussian integer is its product with its conjugate.
N(a+bi)=(a+bi)(a-bi)=a2+b2.
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. Thus a norm cannot be of the form, with integer.
The norm is multiplicative, that is, one has
N(zw)=N(z)N(w),
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, and .
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 and divisor, and produces a quotient and remainder such that
a=bq+r and N(r)<N(b).
a=bq+r and N(r)\le
| N(b) | |
| 2 |
.
To prove this, one may consider the complex number quotient . There are unique integers and such that and, and thus . Taking, one has
a=bq+r,
r=bl(x-m+i(y-n)r),
N(r)\le
| N(b) | |
| 2 |
.
Since the ring of Gaussian integers is a Euclidean domain, is a principal ideal domain, which means that every ideal of is principal. Explicitly, an ideal is a subset of a ring such that every sum of elements of and every product of an element of by an element of belong to . An ideal is principal if it consists of all multiples of a single element, that is, it has the form
\{gx\midx\inG\}.
Every ideal in the ring of the Gaussian integers is principal, because, if one chooses in a nonzero element of minimal norm, for every element of, the remainder of Euclidean division of by belongs also to and has a norm that is smaller than that of ; because of the choice of, this norm is zero, and thus the remainder is also zero. That is, one has, where is the quotient.
For any, the ideal generated by is also generated by any associate of, that is, ; 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.
In 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 has an odd norm, then one of and is odd, and the other is even. Thus has exactly one associate with a real part 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 is one. In fact, as, 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 by the inverse of this unit, one finds an associate that has one as a remainder, when divided by .
If the norm of is even, then either or, where is a positive integer, and is odd. Thus, one chooses the associate of for getting a which fits the choice of the associates for elements of odd norm.
As the Gaussian integers form a principal ideal domain they form also 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 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, with a nonnegative integer) . The other prime numbers are not Gaussian primes, but each is the product of two conjugate Gaussian primes.
A Gaussian integer is a Gaussian prime if and only if either:
In other words, a Gaussian integer is a Gaussian prime if and only if either its norm is a prime number, or it is the product of a unit and a prime number of the form .
It follows that there are three cases for the factorization of a prime number in the Gaussian integers:
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
| e0 | |
| u(1+i) |
| e1 | |
| {p | |
| 1} |
…
| ek | |
| {p | |
| k} |
,
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, while it is with the second choice.
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
x2+cx+d=0,
x2-2ax+a2+b2,
As for any unique factorization domain, a greatest common divisor (gcd) of two Gaussian integers is a Gaussian integer that is a common divisor of and, which has all common divisors of and as divisor. That is (where denotes the divisibility relation),
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 and is a generator of the ideal generated by and (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 of and, the greatest common divisors of and are, and .
There are several ways for computing a greatest common divisor of two Gaussian integers and . When one knows the prime factorizations of and,
a=
| k\prod | |
| i | |
| m |
| \num | |
| {p | |
| m} |
, b=
| n\prod | |
| i | |
| m |
| \mum | |
| {p | |
| m} |
,
\prodm
| λm | |
| {p | |
| m} |
,
λm=min(\num,\mum).
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 by, where is the remainder of the Euclidean division of by, and repeating this operation until getting a zero remainder, that is a pair . This process terminates, because, at each step, the norm of the second Gaussian integer decreases. The resulting is a greatest common divisor, because (at each step) and have the same divisors as and, 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 of the greatest common divisor of and is a common divisor of,, and . When the greatest common divisor of these three integers has few factors, then it is easy to test, for common divisor, all Gaussian integers with a norm dividing .
For example, if, and, one has,, and . As the greatest common divisor of the three norms is 2, the greatest common divisor of and has 1 or 2 as a norm. As a gaussian integer of norm 2 is necessary associated to, and as divides and, then the greatest common divisor is .
If is replaced by its conjugate, then the greatest common divisor of the three norms is 34, the norm of, thus one may guess that the greatest common divisor is, that is, that . In fact, one has .
Given a Gaussian integer, called a modulus, two Gaussian integers are congruent modulo, if their difference is a multiple of, that is if there exists a Gaussian integer such that . In other words, two Gaussian integers are congruent modulo, if their difference belongs to the ideal generated by . This is denoted as .
The congruence modulo 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, or, or simply .
The residue class of a Gaussian integer is the set
\bara:=\left\{z\inZ[i]\midz\equiva\pmod{z0}\right\}
Addition and multiplication are compatible with congruences. This means that and imply and .This defines well-defined operations (that is independent of the choice of representatives) on the residue classes:
\bara+\barb:=\overline{a+b} and \bara ⋅ \barb:=\overline{ab}.
Given a modulus, all elements of a residue class have the same remainder for the Euclidean division by, 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
\begin{align} Vs&=\left\{\left.z0\left(s-\tfrac12+ix\right)\right\vertx\inR\right\} and\\ Ht&=\left\{\left.z0\left(x+i\left(t-\tfrac12\right)\right)\right\vertx\inR\right\}, \end{align}
with and integers (blue lines in the figure). These divide the plane in semi-open squares (where and are integers)
Qmn=\left\{(s+it)z0\left\verts\in\left[m-\tfrac12,m+\tfrac12\right),t\in\left[n-\tfrac12,n+\tfrac12\right)\right.\right\}.
The semi-open intervals that occur in the definition of have been chosen in order that every complex number belong to exactly one square; that is, the squares form a partition of the complex plane. One has
Qmn=(m+in)z0+Q00=\left\{(m+in)z0+z\midz\inQ00\right\}.
This implies that every Gaussian integer is congruent modulo to a unique Gaussian integer in (the green square in the figure), which its remainder for the division by . In other words, every residue class contains exactly one element in .
The Gaussian integers in (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 equals its norm (see below for a proof; similarly, for integers, the number of residue classes modulo is its absolute value).
The residue class ring modulo a Gaussian integer is a field if and only if
z0
If is a decomposed prime or the ramified prime (that is, if its norm 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,). It is thus isomorphic to the field of the integers modulo .
If, on the other hand, is an inert prime (that is, is the square of a prime number, which is congruent to 3 modulo 4), then the residue class field has elements, and it is an extension of degree 2 (unique, up to an isomorphism) of the prime field with elements (the integers modulo).
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 ) and Euler's totient function. The primitive residue class group of a modulus is defined as the subset of its residue classes, which contains all residue classes that are coprime to, i.e. . Obviously, this system builds a multiplicative group. The number of its elements shall be denoted by (analogously to Euler's totient function for integers).
For Gaussian primes it immediately follows that and for arbitrary composite Gaussian integers
z=
| k\prod | |
| i | |
| m |
| \num | |
| {p | |
| m} |
\phi(z)
| =\prod | |
| m(\num>0) |
| \num | |
| l|{p | |
| m} |
r|2\left(1-
| 1{|p | |
| m|{} |
2}\right)=
| 2\prod | |
| |z| | |
| pm|z |
\left(1-
| 1{|p | |
| m|{} |
2}\right)
For all with, it holds that .
The ring of Gaussian integers was introduced by Carl Friedrich Gauss in his second monograph on quartic reciprocity (1832). The theorem of quadratic reciprocity (which he had first succeeded in proving in 1796) relates the solvability of the congruence to that of . Similarly, cubic reciprocity relates the solvability of to that of, and biquadratic (or quartic) reciprocity is a relation between and . 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.
Most of the unsolved problems are related to distribution of Gaussian primes in the plane.
There are also conjectures and unsolved problems about the Gaussian primes. Two of them are: