Proceedings of the 44th Symposium on Ring Theory and ...

Pro<strong>of</strong> <strong>of</strong> (2). We consider <strong>the</strong> case n = p = 3. If f is composite, <strong>the</strong>n f does not
divide t. Thus we may assume f is prime and so r = f (see [7]). f has a primary prime
decomposition f = η¯η in Z[ω] where ω = e 2πi
3 and η = ω(ω − q), (see [6, 8]). In this case,
we set χ is <strong>the</strong> cubic residue character modulo η. Let h(x) be <strong>the</strong> minimal polynomial <strong>of</strong>
θ over Q.
h(x) := x 3 + a 1 x 2 + a 2 x + a 3 = (x − θ 0 )(x − θ 1 )(x − θ 2 ).
where a 1 = −θ 0 − θ 1 − θ 2 = 1. If 3 does not divide I(θ), <strong>the</strong>n h(x) ≡ x 3 − x mod 3 by
Kummer’s <strong>the</strong>orem and our Theorem. This contradicts to a 1 = 1. Thus d(θ) ≡ 0 mod 3.
Using g 1 g 2 = g 1 ḡ 1 = |g 1 | 2 = r, we have
∣ 1 θ 1 θ 2 ∣∣∣∣∣
f = r = −|A 3 | = −(θ 0 + θ 1 + θ 2 )
1 θ 0 θ 1 = θ0 2 + θ1 2 + θ2 2 − a 2 = 1 − 3a 2 .
∣ 1 θ 2 θ 0
Thus we obtain 3a 2 = 1 − f = −q(q + 1). On <strong>the</strong> o<strong>the</strong>r hand, using g 2 = ḡ 1 , f = η¯η and
<strong>the</strong> Stickelberger relation g1 3 = rη = fη (see [6]), we have
∣ g 0 g 1 g 2 ∣∣∣∣∣
−27a 3 = 27θ 0 θ 1 θ 2 = |B 3 | =
g 2 g 0 g 1 = g0 3 + g1 3 + g2 3 − 3g 0 g 1 g 2
∣ g 1 g 2 g 0
= −1 + f(η + ¯η) + 3f = −1 + f(q − 1) + 3f = (q + 1) 3 .
Thus we have 3 3 q 3 a 3 = (−q(q + 1)) 3 = 3 3 a 3 2 and so a 2 + a 3 ≡ a 3 2 − q 3 a 3 = 0 mod 3.
Noting h ′ (θ) ≡ a 2 − θ mod 3 where h ′ (x) is <strong>the</strong> derivation <strong>of</strong> h(x), we obtain
0 ≡ −d(θ) = N L3 /Q(h ′ (θ)) ≡ h(a 2 ) ≡ a 2 − a 2 2 + a 3 ≡ −a 2 2 mod 3.
Thus we have 0 ≡ 3a 2 = −q(q + 1) mod 9.
Remark. Using only <strong>the</strong> quadratic reciprocity law, we can prove
q ≡ −1 mod 8 in case p = 3 and f divides t.
It simplifies <strong>the</strong> pro<strong>of</strong> <strong>of</strong> Proposition 3.2 by Lemma 3.3 on p.172 in <strong>the</strong> paper
K. Dilcher and J. Knauer, On a conjecture <strong>of</strong> Feit and Thompson, pp.169-178 in <strong>the</strong> book,
High primes and misdemeanours, edited by A. van der Poorten, A. Stein, Fields Institute
Communications 41, Amer. Math. Soc., 2004.
We can understand <strong>the</strong>ir pro<strong>of</strong> through <strong>the</strong> next some results in this order :
• Ex. 11 on p.231, and p.103 in <strong>the</strong> book, B. C. Berndt, R.J. Evans, K. S. Williams,
Gauss and Jacobi Sums, Wiley, New York, 1998.
• Pro<strong>of</strong> <strong>of</strong> Theorem 2 on p.139 in <strong>the</strong> paper, R. Hudson and K. S. Williams, Some
new residuacity criteria, Pacific J. <strong>of</strong> Math. 91(1980), 135-143.
• The tables for <strong>the</strong> cyclotomic numbers <strong>of</strong> order 6 and p.68 in <strong>the</strong> paper, A. L.
Whiteman, The cyclotomic numbers <strong>of</strong> order twelve, Acta. Arithmetica 6 (1960),
53-76.
✷
–123–