By Fernando Q. Gouvea, Noriko Yui
The complaints of the 3rd convention of the Canadian quantity concept organization August 18-24, 1991
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Additional resources for Advances in number theory: the proceedings of the Third Conference of the Canadian Number Theory Association, August 18-24, 1991, the Queen's University at Kingston
Suppose that deg f(x) most n distinct roots. = n. Then f has at PROOF . The proof goes by induction on n. For O n = 1 the assertion is trivial. Assume that the lemma is true for polynomials of degree n - 1. If f(x) has no roots in k, we are done. If a. is a root, f(x) = q(xXx - «) + r, where r is a constant. Setting x = a. we see that r = O. ) and deg q(x) = n - 1. If fJ :f. a. )q(f3), which implies that q(fJ) = O. Since by induction q(x) has at 0 most n - 1 distinct roots,f(x) has at most n distinct roots.
PROOF. By Theorem 2' we can assume that n # 2', 12':: 3. Ifn is not of the given . form, it is easy to see that n can be written as a product m 1m2' where (m l, m2) = 1 and ml, m2 > 2. We then have that ¢(md and ¢(m2) are both even and that U(7L/n7L);:::: U(7L/m l7L) x U(7L/m 27L) . Both U(7L/m 17L) and U(7L/m 27L) have elements of order 2, but this shows that U(7L/n7L) is not cyclic since a 45 *2 nth Power Residues cyclic group contains at most one element of order 2. Thus n does not possess primitive roots.
2. (a) (b) a = 5 iff a == b (m). a#-5 iffa n 5 is empty. (c) There are precisely m distinct congruence classes modulo m. PROOF. 5 = a, then a E a = 5. Thus a == b (m). Conversely, if a == b (m), then a E 5. If c == a (m), then c == b (m), which shows a ~ 5. Since a == b (m) implies that b == a (m), we also have 5 ~ a. Therefore a = 5. (a) If (b) Clearly, if an 5 is empty, then a#- 5. We shall show that an 5 not empty implies that a = 5. Let c E an 5. Then c == a (m) and c == b (m). It follows that a == b (m) and so by part (a) we have a = 5.
Advances in number theory: the proceedings of the Third Conference of the Canadian Number Theory Association, August 18-24, 1991, the Queen's University at Kingston by Fernando Q. Gouvea, Noriko Yui