In algebraic number theory Eisenstein's reciprocity law is a reciprocity law that extends the law of quadratic reciprocity and the cubic reciprocity law to residues of higher powers. It is one of the earliest and simplest of the higher reciprocity laws, and is a consequence of several later and stronger reciprocity laws such as the Artin reciprocity law. It was introduced by , though Jacobi had previously announced (without proof) a similar result for the special cases of 5th, 8th and 12th powers in 1839.
Let be an integer, and let be the ring of integers of the m-th cyclotomic field where is a
primitive m-th root of unity.
The numbers are units in (There are other units as well.)
A number is called primary if it is not a unit, is relatively prime to , and is congruent to a rational (i.e. in ) integer
The following lemma shows that primary numbers in are analogous to positive integers in
Suppose that and that both and are relatively prime to Then
There is an integer making primary. This integer is unique
if and are primary then is primary, provided that is coprime with .
if and are primary then is primary.
is primary.
The significance of
which appears in the definition is most easily seen when
is a prime. In that case
Furthermore, the prime ideal
of
is totally ramified in
and the ideal
is prime of degree 1.
Power residue symbol
For the m-th power residue symbol for is either zero or an m-th root of unity:
It is the m-th power version of the classical (quadratic, m = 2) Jacobi symbol (assuming and are relatively prime):
If and then
If then is not an m-th power
If then may or may not be an m-th power
Let be an odd prime and an integer relatively prime to Then
Let be primary (and therefore relatively prime to ), and assume that is also relatively prime to . Then
The theorem is a consequence of the Stickelberger relation.
gives a historical discussion of some early reciprocity laws, including a proof of Eisenstein's law using Gauss and Jacobi sums that is based on Eisenstein's original proof.
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In algebraic number theory Eisenstein's reciprocity law is a reciprocity law that extends the law of quadratic reciprocity and the cubic reciprocity law to residues of higher powers. It is one of the earliest and simplest of the higher reciprocity laws, and is a consequence of several later and stronger reciprocity laws such as the Artin reciprocity law. It was introduced by , though Jacobi had previously announced (without proof) a similar result for the special cases of 5th, 8th and 12th powers in 1839.
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: Let p and q be distinct odd prime numbers, and define the Legendre symbol as: Then: This law, together with its supplements, allows the easy calculation of any Legendre symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form for an odd prime ; that is, to determine the "perfect squares" modulo .
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.