Symbol (number theory)

Last updated February 26, 2019

In number theory, a symbol is any of many different generalizations of the Legendre symbol. This article describes the relations between these various generalizations.

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of objects made out of integers or defined as generalizations of the integers.

In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number p: its value at a (nonzero) quadratic residue mod p is 1 and at a non-quadratic residue (non-residue) is −1. Its value at zero is 0.

The symbols below are arranged roughly in order of the date they were introduced, which is usually (but not always) in order of increasing generality.

Legendre symbol $\left({\frac {a}{p}}\right)$ defined for p a prime, a an integer, and takes values 0, 1, or −1.
Jacobi symbol $\left({\frac {a}{b}}\right)$ defined for b a positive odd integer, a an integer, and takes values 0, 1, or −1. An extension of the Legendre symbol to more general values of b.
Kronecker symbol $\left({\frac {a}{b}}\right)$ defined for b any integer, a an integer, and takes values 0, 1, or −1. An extension of the Jacobi and Legendre symbols to more general values of b.
Power residue symbol $\left({\frac {a}{b}}\right)=\left({\frac {a}{b}}\right)_{m}$ is defined for a in some global field containing the mth roots of 1 ( for some m), b a fractional ideal of K built from prime ideals coprime to m. The symbol takes values in the m roots of 1. When m = 2 and the global field is the rationals this is more or less the same as the Jacobi symbol.
Hilbert symbol The local Hilbert symbol (a,b) = is defined for a and b in some local field containing the m roots of 1 (for some m) and takes values in the m roots of 1. The power residue symbol can be written in terms of the Hilbert symbol. The global Hilbert symbol $(a,b)_{p}=\left({\frac {a,b}{p}}\right)=\left({\frac {a,b}{p}}\right)_{m}$ is defined for a and b in some global field K, for p a finite or infinite place of K, and is equal to the local Hilbert symbol in the completion of K at the place p.
Artin symbol The local Artin symbol or norm residue symbol $\theta _{L/K}(\alpha )=(\alpha ,L/K)=\left({\frac {L/K}{\alpha }}\right)$ is defined for L a finite extension of the local field K, α an element of K, and takes values in the abelianization of the Galois group Gal(L/K). The global Artin symbol $\psi _{L/K}(\alpha )=(\alpha ,L/K)=\left({\frac {L/K}{\alpha }}\right)=((L/K)/\alpha )$ is defined for α in a ray class group or idele (class) group of a global field K, and takes values in the abelianization of Gal(L/K) for L an abelian extension of K. When α is in the idele group the symbol is sometimes called a Chevalley symbol or Artin–Chevalley symbol. The local Hilbert symbol of K can be written in terms of the Artin symbol for Kummer extensions L/K, where the roots of unity can be identified with elements of the Galois group.
The Frobenius symbol $[(L/K)/P]=\left[{\frac {L/K}{P}}\right]$ is the same as the Frobenius element of the prime P of the Galois extension L of K.
"Chevalley symbol" has several slightly different meanings. It is sometimes used for the Artin symbol for ideles. A variation of this is the Chevalley symbol $\left({\frac {a,\chi }{p}}\right)$ for p a prime ideal of K, a an element of K, and χ a homomorphism of the Galois group of K to R/Z. The value of the symbol is then the value of the character χ on the usual Artin symbol.
Norm residue symbol This name is for several different closely related symbols, such as the Artin symbol or the Hilbert symbol or Hasse's norm residue symbol. The Hasse norm residue symbol $((\alpha ,L/K)/p)=\left({\frac {\alpha ,L/K}{p}}\right)$ is defined if p is a place of K and α an element of K. It is essentially the same as the local Artin symbol for the localization of K at p. The Hilbert symbol is a special case of it in the case of Kummer extensions.
Steinberg symbol (a,b). This is a generalization of the local Hilbert symbol to arbitrary fields F. The numbers a and b are elements of F, and the symbol (a,b) takes values in the second K-group of F.
Galois symbol A sort of generalization of the Steinberg symbol to higher algebraic K-theory. It takes a Milnor K-group to an étale cohomology group.

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 turn are important in cryptography.

In number theory, the Kronecker symbol, written as $or, is a generalization of the Jacobi symbol to all integers . It was introduced by Leopold Kronecker.$

In algebraic number theory the n-th power residue symbol is a generalization of the (quadratic) Legendre symbol to n-th powers. These symbols are used in the statement and proof of cubic, quartic, Eisenstein, and related higher reciprocity laws.

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In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is:

Algebraic number theory major branch of number theory

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.

In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that:

In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of local fields and "global fields" such as number fields and function fields of curves over finite fields in terms of abelian topological groups associated to the fields. It also studies various arithmetic properties of such abelian extensions. Class field theory includes global class field theory and local class field theory.

In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity.

In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield.

In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields is an important tool in number theory.

In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's last theorem. The main statements do not depend on the nature of the field – apart from its characteristic, which should not divide the integer n – and therefore belong to abstract algebra. The theory of cyclic extensions of the field K when the characteristic of K does divide n is called Artin–Schreier theory.

In abstract algebra, Hilbert's Theorem 90 is an important result on cyclic extensions of fields that leads to Kummer theory. In its most basic form, it states that if L/K is a cyclic extension of fields with Galois group G = Gal(L/K) generated by an element $and if is an element of L of relative norm 1, then there exists in L such that$

Gauss's lemma in number theory gives a condition for an integer to be a quadratic residue. Although it is not useful computationally, it has theoretical significance, being involved in some proofs of quadratic reciprocity.

The Artin reciprocity law, which was established by Emil Artin in a series of papers, is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.

In mathematics, the Hilbert symbol or norm-residue symbol is a function from K^× × K^× to the group of nth roots of unity in a local field K such as the fields of reals or p-adic numbers. It is related to reciprocity laws, and can be defined in terms of the Artin symbol of local class field theory. The Hilbert symbol was introduced by David Hilbert in his Zahlbericht, with the slight difference that he defined it for elements of global fields rather than for the larger local fields.

Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x³ ≡ p (mod q) 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 ring of Eisenstein integers, both coprime to 3, the congruence x³ ≡ p is solvable if and only if x³ ≡ q is solvable.

Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x⁴ ≡ p is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the congruence x⁴ ≡ p to that of x⁴ ≡ q.

In mathematics, an explicit reciprocity law is a formula for the Hilbert symbol of a local field. The name "explicit reciprocity law" refers to the fact that the Hilbert symbols of local fields appear in Hilbert's reciprocity law for the power residue symbol. The definitions of the Hilbert symbol are usually rather roundabout and can be hard to use directly in explicit examples, and the explicit reciprocity laws give more explicit expressions for the Hilbert symbol that are sometimes easier to use.

In mathematics, a ray class field is an abelian extension of a global field associated with a ray class group of ideal classes or idele classes. Every finite abelian extension of a number field is contained in one of its ray class fields.

References

Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.

Jürgen Neukirch was a German mathematician known for his work on algebraic number theory.

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Symbol (number theory)

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