Cubic reciprocity

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In mathematics, cubic reciprocity refers to various results connecting the solvability of two related cubic equations in modular arithmetic. It is a generalisation of the concept of quadratic reciprocity.

Algebraic setting

The law of cubic reciprocity is most naturally expressed in terms of the Eisenstein integers, that is, the ring E of complex numbers of the form

where and a and b are integers and

is a complex cube root of unity.

If is a prime element of E of norm P and is an element coprime to , we define the cubic residue symbol to be the cube root of unity (power of ) satisfying

We further define a primary prime to be one which is congruent to -1 modulo 3. Then for distinct primary primes and the law of cubic reciprocity is simply

with the supplementary laws for the units and for the prime of norm 3 that if then


References

  • David A. Cox, Primes of the form , Wiley, 1989, ISBN 0-471-50654-0.
  • K. Ireland and M. Rosen, A classical introduction to modern number theory, 2nd ed, Graduate Texts in Mathematics 84, Springer-Verlag, 1990.
  • Franz Lemmermeyer, Reciprocity laws: From Euler to Eisenstein, Springer Verlag, 2000, ISBN 3-540-66957-4.

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