14 Gaussians

If we adjoin a square root of -1, usually denoted by i, to the field of rationals we obtain a field that is an extension of degree 2. This field is called the Gaussian rationals and its ring of integers is called the Gaussian integers, because C.F. Gauss was the first to study them.

In GAP Gaussian rationals are written in the form a + b*E(4), where a and b are rationals, because E(4) is GAP's name for i. Because 1 and i form an integral base the Gaussian integers are written in the form a + b*E(4), where a and b are integers.

The first sections in this chapter describe the operations applicable to Operations for Gaussians).

The next sections describe the functions that test whether an object is a Gaussian rational or integer (see IsGaussRat and IsGaussInt).

The GAP object GaussianRationals is the field domain of all Gaussian rationals, and the object GaussianIntegers is the ring domain of all Gaussian integers. All set theoretic functions are applicable to those two domains (see chapter Domains and Set Functions for Gaussians).

The Gaussian rationals form a field so all field functions, e.g., Norm, are applicable to the domain GaussianRationals and its elements (see chapter Fields and Field Functions for Gaussian Rationals).

The Gaussian integers form a Euclidean ring so all ring functions, e.g., Factors, are applicable to GaussianIntegers and its elements (see chapter Rings, Ring Functions for Gaussian Integers, and TwoSquares).

The field of Gaussian rationals is just a special case of cyclotomic fields, so everything that applies to those fields also applies to it (see chapters Cyclotomics and Subfields of Cyclotomic Fields).

All functions are in the library file LIBNAME/"gaussian.g".

Subsections

  1. Comparisons of Gaussians
  2. Operations for Gaussians
  3. IsGaussRat
  4. IsGaussInt
  5. Set Functions for Gaussians
  6. Field Functions for Gaussian Rationals
  7. Ring Functions for Gaussian Integers
  8. TwoSquares

14.1 Comparisons of Gaussians

x = y
x < y

The equality operator evaluates to true if the two Gaussians x and y are equal, and to false otherwise. The inequality operator < evaluates to true if the two Gaussians x and y are not equal, and to false otherwise. It is also possible to compare a Gaussian with an object of another type, of course they are never equal.

Two Gaussians a + b*E(4) and c + d*E(4) are considered equal if a = c and b = d.

    gap> 1 + E(4) = 2 / (1 - E(4));
    true
    gap> 1 + E(4) = 1 - E(4);
    false
    gap> 1 + E(4) = E(6);
    false 

x < y
x <= y
x y
x = y

The operators <, <=, , and = evaluate to true if the Gaussian x is less than, less than or equal to, greater than, and greater than or equal to the Gaussian y, and to false otherwise. Gaussians can also be compared to objects of other types, they are Comparisons of Cyclotomics).

A Gaussian a + b*E(4) is considered less than another Gaussian c + d*E(4) if a is less than c, or if a is equal to c and b is less than d.

    gap> 1 + E(4) < 2 + E(4);
    true
    gap> 1 + E(4) < 1 - E(4);
    false
    gap> 1 + E(4) < 1/2;
    false 

14.2 Operations for Gaussians

x + y
x - y
x * y
x / y

The operators +, -, *, and / evaluate to the sum, difference, product, and quotient of the two Gaussians x and y. Of course either operand may also be an ordinary rational (see Rationals), because the rationals are embedded into the Gaussian rationals. On the other hand the Gaussian rationals are embedded into other cyclotomic fields, so either operand may also be a cyclotomic (see Cyclotomics). Division by 0 is as usual an error.

x ^ n

The operator ^ evaluates to the n-th power of the Gaussian rational x. If n is positive, the power is defined as the n-fold product x*x*...x; if n is negative, the power is defined as (1/x)^(-n); and if n is zero, the power is 1, even if x is 0.

    gap> (1 + E(4)) * (E(4) - 1);
    -2 

14.3 IsGaussRat

IsGaussRat( obj )

IsGaussRat returns true if obj, which may be an object of arbitrary type, is a Gaussian rational and false otherwise. Will signal an error if obj is an unbound variable.

    gap> IsGaussRat( 1/2 );
    true
    gap> IsGaussRat( E(4) );
    true
    gap> IsGaussRat( E(6) );
    false
    gap> IsGaussRat( true );
    false 

IsGaussInt can be used to test whether an object is a Gaussian integer (see IsGaussInt).

14.4 IsGaussInt

IsGaussInt( obj )

IsGaussInt returns true if obj, which may be an object of arbitrary type, is a Gaussian integer, and false otherwise. Will signal an error if obj is an unbound variable.

    gap> IsGaussInt( 1 );
    true
    gap> IsGaussInt( E(4) );
    true
    gap> IsGaussInt( 1/2 + 1/2*E(4) );
    false
    gap> IsGaussInt( E(6) );
    false 

IsGaussRat can be used to test whether an object is a Gaussian rational (see IsGaussRat).

14.5 Set Functions for Gaussians

As already mentioned in the introduction of this chapter the objects GaussianRationals and GaussianIntegers are the domains of Gaussian rationals and integers respectively. All set theoretic functions, i.e., Size and Intersection, are applicable to these domains and their elements (see chapter Domains). There does not seem to be an important use of this however. All functions not mentioned here are not treated specially, i.e., they are implemented by the default function mentioned in the respective section.

in

The membership test for Gaussian rationals is implemented via IsGaussRat (IsGaussRat). The membership test for Gaussian integers is implemented via IsGaussInt (see IsGaussInt).

Random

A random Gaussian rational a + b*E(4) is computed by combining two random rationals a and b (see Set Functions for Rationals). Likewise a random Gaussian integer a + b*E(4) is computed by Set Functions for Integers).

    gap> Size( GaussianRationals );
    "infinity"
    gap> Intersection( GaussianIntegers, [1,1/2,E(4),-E(6),E(4)/3] );
    [ 1, E(4) ] 

14.6 Field Functions for Gaussian Rationals

As already mentioned in the introduction of this chapter, the domain of Gaussian rationals is a field. Therefore all field functions are applicable to this domain and its elements (see chapter Fields). This section gives further comments on the definitions and implementations of those functions for the the Gaussian rationals. All functions not mentioned here are not treated specially, i.e., they are implemented by the default function mentioned in the respective section.

Conjugates

The field of Gaussian rationals is an extension of degree 2 of the rationals, its prime field. Therefore there is one further conjugate of every element a + b*E(4), namely a - b*E(4).

Norm, Trace

According to the definition of conjugates above, the norm of a Gaussian rational a + b*E(4) is a^2 + b^2 and the trace is 2*a.

14.7 Ring Functions for Gaussian Integers

As already mentioned in the introduction to this chapter, the ring of Gaussian integers is a Euclidean ring. Therefore all ring functions are applicable to this ring and its elements (see chapter Rings). This section gives further comments on the definitions and implementations of those functions for the Gaussian integers. All functions not mentioned here are not treated specially, i.e., they are implemented by the default function mentioned in the respective section.

IsUnit, Units, IsAssociated, Associates

The units of GaussianIntegers are [ 1, E(4), -1, -E(4) ].

StandardAssociate

The standard associate of a Gaussian integer x is the associated element y of x that lies in the first quadrant of the complex plane. That is y is that element from x * [1,-1,E(4),-E(4)] that has positive real part and nonnegative imaginary part.

EuclideanDegree

The Euclidean degree of a Gaussian integer x is the product of x and its complex conjugate.

EuclideanRemainder

Define the integer part i of the quotient of x and y as the point of the lattice spanned by 1 and E(4) that lies next to the rational quotient of x and y, rounding towards the origin if there are several such points. Then EuclideanRemainder( x, y ) is defined as x - i * y. With this definition the ordinary Euclidean algorithm for the greatest common divisor works, whereas it does not work if you always round towards the origin.

EuclideanQuotient

The Euclidean quotient of two Gaussian integers x and y is the quotient of w and y, where w is the difference between x and the Euclidean remainder of x and y.

QuotientRemainder

QuotientRemainder uses EuclideanRemainder and EuclideanQuotient.

IsPrime, IsIrreducible

Since the Gaussian integers are a Euclidean ring, primes and irreducibles are equivalent. The primes are the elements 1 + E(4) and 1 - E(4) of norm 2, the elements a + b*E(4) and a - b*E(4) of norm p = a^2 + b^2 with p a rational prime congruent to 1 mod 4, and the elements p of norm p^2 with p a rational prime congruent to 3 mod 4.

Factors

The list returned by Factors is sorted according to the norms of the primes, and among those of equal norm with respect to <. All elements in the list are standard associates, except the first, which is multiplied by a unit as necessary.

The above characterization already shows how one can factor a Gaussian integer. First compute the norm of the element, factor this norm over the rational integers and then split 2 and the primes congruent to 1 mod 4 with TwoSquares (see TwoSquares).

    gap> Factors( GaussianIntegers, 30 );
    [ -1-E(4), 1+E(4), 3, 1+2*E(4), 2+E(4) ] 

14.8 TwoSquares

TwoSquares( n )

TwoSquares returns a list of two integers x<=y such that the sum of the squares of x and y is equal to the nonnegative integer n, i.e., n = x^2+y^2. If no such representation exists TwoSquares will return false. TwoSquares will return a representation for which the gcd of x and y is as small as possible. If there are several such representations, it is not specified which one TwoSquares returns.

Let a be the product of all maximal powers of primes of the form 4k+3 dividing n. A representation of n as a sum of two squares exists if and only if a is a perfect square. Let b be the maximal power of 2 dividing n, or its half, whichever is a perfect square. Then the minimal possible gcd of x and y is the square root c of a b. The number of different minimal representations with x<=y is 2^{l-1}, where l is the number of different prime factors of the form 4k+1 of n.

    gap> TwoSquares( 5 );
    [ 1, 2 ]
    gap> TwoSquares( 11 );
    false        # no representation exists
    gap> TwoSquares( 16 );
    [ 0, 4 ]
    gap> TwoSquares( 45 );
    [ 3, 6 ]        # 3 is the minimal possible gcd because 9 divides 45
    gap> TwoSquares( 125 );
    [ 2, 11 ]        # not [ 5, 10 ] because this has not minimal gcd
    gap> TwoSquares( 13*17 );
    [ 5, 14 ]        # [10,11] would be the other possible representation
    gap> TwoSquares( 848654483879497562821 );
    [ 6305894639, 28440994650 ]        # 848654483879497562821 is prime 

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GAP 3.4.4
April 1997