Question

Find the gcd of 1008, -357 and express the gcd in the form m(1008) + ????(−357)

Answer 1

Answer:

Explanation:

Let m > 2 be an integer and ζm an m-th primitive root of 1. For each prime

q ≡ 1 mod 2m let ζq be a q-th primitive root of 1, s = sq a primitive root modulo

q and f = fq = (q − 1)/m (we will assume that f is even for simplicity). Let S

be the set of all primes q ≡ 1 mod 2m. Given q ∈ S, define the Jacobi sums Ja,b,

0 ≤ a, b ≤ m−1, and the Gaussian periods ηi, 0 ≤ i ≤ m−1, of degree m in Q(ζq),

by

Ja,b = −

Xq−1

k=2

ζ a inds(k)+b inds(1−k) m ,

where inds(k) is the least nonnegative integer such that s inds(k) ≡ k mod q, and

ηi =

f

X−1

j=0

ζsi+mj

q .

Define Pq(x) = Qm−1

i=0 (x−ηi), the irreducible polynomial, over Q, of the periods ηi.

In this article we study the numbers Ja,b, and use them to construct large families

of polynomials Pq(x), q ∈ P, where P is a subset of S. In principle the method

shown here would allow us to construct a finite number of such families, whose

indices put together include all the primes in S.

This research originated from a problem indicated to me by Ren´e Schoof. The

first part of the problem was to find, for m = 7, or m = 9, or m = 12, families of

Received by the editor September 15, 1998 and, in revised form, January 19, 2000.

2000 Mathematics Subject Classification. Primary 11R18, 11R21, 11T22.

This work was supported in part by grants from NSERC and FCAR.