On Values of Cyclotomic Polynomials. V

In this paper, we present three results on cyclotomic polynomials. First, we present results about factorization of cyclotomic polynomials over arbitrary fields K. It is well known in cases such that a field K is the rational number field Q or a finite field F q (see [3, 4]). Using irreducibility of cyclotomic polynomials over Q, we can see that there are only finite elements of finite orders in a number field. On the other hand, we should correct some mistakes in [2, Corollary 1]. This mistake have no influence about another results in [2]. Finaly, we state about relations between Fibonacci polynomials and cyclotomic polynomials. This idea is due to K. Kuwano who stated this in his book [1] written in Japanese. 1. Factorizations of cyclotomic polynomials over fields The next theorem shows that irreducible factors of a cyclotomic polynomial Φn(x) over an arbi-trary field have the same degree. Theorem 1. Let K be a field. Then every irreducible factor f(x) of Φn(x) in K[x] has the same degree. More precisely, let L be the minimal splitting field of Φn(x) over a field K of characteristic p ≥ 0. Then we obtain that L is Galois over K, the Galois group G of L over K is a subgroup of the unit group of Z/mZ, where m = n in case p = 0 and n = pem with (m, p) = 1 in case p> 0, and deg f(x) = |G | = [L: K]. Proof. Let f(x) be a monic irreducible factor of Φn(x) in K[x] and let α ∈ L be a root of f(x). Then n = pem by [2, Theorem 1] where m is the order of α in L and m is not divided by p. Thus, we can see from the equation xm − 1 =∏d|m Φd(x) tha

On Group Structures of Some Special Elliptic Curves

The purpose of this paper is to determine the structures of groups of rational points on elliptic curves of form y2 = x3 − px where p is a Fermat or Mersenne prime. Let E be an elliptic curve y2 = x3 − px where p is a prime and let Γ be the set of rational points in E. Then Γ has an abelian group structure. Mordell-Weil theorem states that Γ is finitely generated. Thus we can set Γ = F ⊕ T where F and T are the free part and the torsion part of Γ, respectively. Let β be a natural group homomorphism from Q × to Q × = Q×/Q×2 and let α be the group homomorphism from Γ to Q × defined by α(P) = 1 for P = Oβ(−p) for P = 0 β(x) for x 6 = 0 where P = (x, y) ∈ Γ, 0 = (0, 0) is the origin and O is the point at infinity. We consider an elliptic curve E ̄ : y2 = x3 + 4px corresponding to E and we similarly define α ̄ from Γ ̄ to Q×, namely, ᾱ(P) = 1 for P = Oβ(p) for P = 0 β(x) for x 6 = 0 where P = (x, y) ∈ Γ̄. The rank r of the free part F of Γ is computed from the formul

A chemical biology approach reveals an opposite action between thermospermine and auxin in xylem development in Arabidopsis thaliana

Thermospermine, a structural isomer of spermine, is produced through the action of ACAULIS5 (ACL5) and suppresses xylem differentiation in Arabidopsis thaliana. To elucidate the molecular basis of the function of thermospermine, we screened chemical libraries for compounds that can modulate xylem differentiation in the acl5 mutant, which is deficient in thermospermine and shows a severe dwarf phenotype associated with excessive proliferation of xylem vessels. We found that the isooctyl ester of a synthetic auxin, 2,4-D, remarkably enhanced xylem vessel differentiation in acl5 seedlings. 2,4-D, 2,4-D analogs and IAA analogs, including 4-chloro IAA (4-Cl-IAA) and IAA ethyl ester, also enhanced xylem vessel formation, while IAA alone had little or no obvious effect on xylem differentiation. These effects of auxin analogs were observed only in the acl5 mutant but not in the wild type, and were suppressed by the anti-auxin, p-chlorophenoxyisobutyric acid (PCIB) and alpha-(phenyl ethyl-2-one)-IAA (PEO-IAA), and also by thermospermine. Furthermore, the suppressor of acaulis51-d (sac51-d) mutation, which causes SAC51 overexpression in the absence of thermospermine and suppresses the dwarf phenotype of acl5, also suppressed the effect of auxin analogs in acl5. These results suggest that the auxin signaling that promotes xylem differentiation is normally limited by SAC51-mediated thermospermine signaling but can be continually stimulated by exogenous auxin analogs in the absence of thermospermine. The opposite action between thermospermine and auxin may fine-tune the timing and spatial pattern of xylem differentiation