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In mathematics, a character sum is a sum () of values of a Dirichlet character χ modulo N, taken over a given range of values of n.Such sums are basic in a number of questions, for example in the distribution of quadratic residues, and in particular in the classical question of finding an upper bound for the least quadratic non-residue modulo N.
If a character of the finite group G is restricted to a subgroup H, then the result is also a character of H. Every character value χ(g) is a sum of n m-th roots of unity, where n is the degree (that is, the dimension of the associated vector space) of the representation with character χ and m is the order of g.
An imprimitive character is induced by the character for the smallest modulus: , is induced from , and , and , are induced from ,. A related phenomenon can happen with a character mod the product of primes; its nonzero values may be periodic with a smaller period.
In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total.Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
A multiplicative character (or linear character, or simply character) on a group G is a group homomorphism from G to the multiplicative group of a field , usually the field of complex numbers. If G is any group, then the set Ch( G ) of these morphisms forms an abelian group under pointwise multiplication.
In mathematics, a Jacobi sum is a type of character sum formed with Dirichlet characters. Simple examples would be Jacobi sums J ( χ , ψ ) for Dirichlet characters χ , ψ modulo a prime number p , defined by
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Algebraic characters are defined for locally-finite weight modules and are additive, i.e. the character of a direct sum of modules is the sum of their characters.On the other hand, although one can define multiplication of the formal exponents by the formula = + and extend it to their finite linear combinations by linearity, this does not make into a ring, because of the possibility of formal ...