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