In number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function

Exponential function

In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

 with coefficients given by a quadratic character; for a general character, one obtains a more general Gauss sum

Gauss sum

In mathematics, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typicallyG := G= \sum \chi\cdot \psi...

. These objects are named after Carl Friedrich Gauss

Carl Friedrich Gauss

Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

, who studied them extensively and applied them to quadratic

Quadratic reciprocity

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers...

, cubic

Cubic reciprocity

Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x3 ≡ p  is solvable; the word "reciprocity" comes from the form of the main theorem, which states that if p and q are primary numbers in the...

, and biquadratic reciprocity laws.

Definition

Let p be an odd prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

 and a an integer. Then the Gauss sum mod p, g(a;p), is the following sum of the pth roots of unity

Root of unity

In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...

:

If a is not divisible by p, an alternative expression for the Gauss sum (with the same value) is

Here is the Legendre symbol

Legendre symbol

In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo a prime number p: its value on a quadratic residue mod p is 1 and on a quadratic non-residue is −1....

, which is a quadratic character mod p. An analogous formula with a general character χ in place of the Legendre symbol defines the Gauss sum

Gauss sum

In mathematics, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typicallyG := G= \sum \chi\cdot \psi...

 G(χ).

Properties

• The value of the Gauss sum is an algebraic integer
  Algebraic integer
  In number theory, an algebraic integer is a complex number that is a root of some monic polynomial with coefficients in . The set of all algebraic integers is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A...
  
   in the pth cyclotomic field
  Cyclotomic field
  In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
  
   Q(ζ_p).

• The evaluation of the Gauss sum can be reduced to the case a = 1:

• The exact value of the Gauss sum, computed by Gauss, is given by the formula

The fact that was easy to prove and led to one of Gauss's proofs of quadratic reciprocity

Proofs of quadratic reciprocity

In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusual number of proofs. Several hundred proofs of the law of quadratic reciprocity have been found.-Proofs that are accessible:...

. However, the determination of the sign of the Gauss sum turned out to be considerably more difficult: Gauss could only establish it after several years' work. Later, Dirichlet, Kronecker, Schur

Issai Schur

Issai Schur was a mathematician who worked in Germany for most of his life. He studied at Berlin...

 and other mathematicians found different proofs.

Generalized quadratic Gauss sums

Let a,b,c be natural numbers. The generalized Gauss sum G(a,b,c) is defined by


where e(x) is the exponential function exp(2πix). The classical Gauss sum is the sum .

Properties

• The Gauss sum G(a,b,c) depends only on the residue class of a,b modulo c.

• Gauss sums are multiplicative
  Multiplicative function
  In number theory, a multiplicative function is an arithmetic function f of the positive integer n with the property that f = 1 and whenevera and b are coprime, then...
  
  , i.e. given natural numbers a, b, c and d with gcd
  Greatest common divisor
  In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
  
  (c,d) =1 one has

G(a,b,cd)=G(ac,b,d)G(ad,b,c).

This is a direct consequence of the Chinese remainder theorem

Chinese remainder theorem

The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.In its most basic form it concerned with determining n, given the remainders generated by division of n by several numbers...

.

• One has G(a,b,c)=0 if gcd(a,c)>1 except if gcd(a,c) divides b in which case one has
  
  Thus in the evaluation of quadratic Gauss sums one may always assume gcd(a,c)=1.
  
  • Let a,b and c be integers with and ac+b even. One has the following analogue of the quadratic reciprocity
    Quadratic reciprocity
    In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers...
    
     law for (even more general) Gauss sums
    
    • Define for every odd integer m.
    
    
    The values of Gauss sums with b=0 and gcd(a,c)=1 are explicitly given by
    
    
    Here is the Jacobi symbol
    Jacobi symbol
    The Jacobi symbol is a generalization of the Legendre symbol. Introduced by Jacobi in 1837, it is of theoretical interest in modular arithmetic and other branches of number theory, but its main use is in computational number theory, especially primality testing and integer factorization; these in...
    
    . This is the famous formula of Carl Friedrich Gauß.
    
    • For b>0 the Gauss sums can easily be computed by completing the square
      Completing the square
      In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the formax^2 + bx + c\,\!to the formIn this context, "constant" means not depending on x. The expression inside the parenthesis is of the form ...
      
       in most cases. This fails however in some cases (for example c even and b odd) which can be computed relatively easy by other means. For example if c is odd and gcd(a,c)=1 one has
    
    
    
    where is some number with . As another example, if 4 divides c and b is odd and as always gcd(a,c)=1 then G(a,b,c)=0. This can, for example, be proven as follows: Because of the multiplicative property of Gauss sums we only have to show that if n>1 and a,b are odd with gcd(a,c)=1. If b is odd then is even for all . By Hensel's lemma
    Hensel's lemma
    In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a polynomial equation has a simple root modulo a prime number , then this root corresponds to a unique root of the same equation modulo any higher power...
    
    , for every q, the equation has at most two solutions in . Because of a counting argument runs through all even residue classes modulo c exactly two times. The geometric sum formula then shows that .
    
    • If c is odd and squarefree
      Square-free integer
      In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32...
      
      and gcd(a,c)=1 then
    
    
    
    If c is not squarefree then the right side vanishes while the left side does not. Often the right sum is also called a quadratic Gauss sum.
    
    • Another useful formula is
    
    

    G(n,p^k)=pG(n,p^k-2)

    
    
    if k≥2 and p is an odd prime number or if k≥4 and p=2.
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