cubic reciprocity law
In a ring , a cubic residue is just a value of the function for some invertible element of the ring. Cubic residues display a reciprocity phenomenon similar to that seen with quadratic residues. But we need some preparation in order to state the cubic reciprocity law.
Whereas has only two units (meaning invertible elements), namely , has six, namely all the sixth roots of 1:
is a principal ring, hence has unique factorization. Let us call “irreducible” if the condition implies that or , but not both, is a unit. It turns out that the irreducible elements of are (up to multiplication by units):
– the number , which has norm 3. We will denote it by .
– complex numbers where is a prime in and .
For example, is a prime in because its norm, 7, is prime in and is 1 mod 3; but 7 is not a prime in .
Now we need some convention whereby at most one of any six associates is called a prime. By convention, the following numbers are nominated:
– the number .
– rational primes (rather than their negative or complex associates).
– complex numbers where is prime in and
One can verify that this selection exists and is unambigous.
Next, we seek a three-valued function analogous to the two-valued quadratic residue character . Let be a prime in , with . If is any element of such that , then
Since is a multiple of 3, we can define a function
is a character, called the cubic residue character mod . We have if and only if is a nonzero cube mod . (Compare Euler’s criterion.)
At last we can state this famous result of Eisenstein and Jacobi:
Theorem (Cubic Reciprocity Law): If and are any two distinct primes in , neither of them , then
The quadratic reciprocity law has two “supplements” which describe and . Likewise the cubic law has this supplement, due to Eisenstein:
Theorem: For any prime in , other than ,
Remarks: Some writers refer to our “irreducible” elements as “primes” in ; what we have called primes, they call “primary primes”.
The quadratic reciprocity law would take a simpler form if we were to make a different convention on what is a prime in , a convention similar to the one in : a prime in is either 2 or an irreducible element of such that . The primes would then be 2, -3, 5, -7, -11, 13, …and the QRL would say simply
for any two distinct odd primes and .
|Title||cubic reciprocity law|
|Date of creation||2013-03-22 13:41:26|
|Last modified on||2013-03-22 13:41:26|
|Last modified by||mathcam (2727)|
|Defines||cubic residue character|