Iftikhar Burhanuddin on Mon, 25 Sep 2006 06:07:38 +0200

 Quartic residue symbol

Hi,

I'm in need of some gp code which will compute the quartic/biquadratic
residue symbol. The symbol is defined as follows [Ireland-Rosen page 122]:

Let Z[i] be the ring of Gaussian integers and let pi be an irreducible in
Z[i]. Let N(pi) denote the size of finite field Z[i]/pi Z[i]. Let T(pi) =
(N(pi)-1)/4.

If a \in Z[i], such that pi does not divide a, and (pi) not equal to
(1+i), there exists a unique integer j, 0 <= j <=3 such that
a^T(pi) congruent to i^j (pi).

The quartic residue symbol for a and pi is defined as (a/pi)_4 := i^j

The following script factors Gaussian integers into Gaussian primes and
hence I have a way of generating the irreducibles.

http://www.research.att.com/~njas/sequences/a078458.txt

Apart from setting up nfinit(y^2+1) and don't know how to proceed with the
arithmetic. Any help will be greatly appreciated!

Regards,
Ifti