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 Rings over Semi-primary Rings II

Article信州大学理学部紀要 6(2): 97-99(1972)departmental bulletin pape

Note on the Endomorphism Ring

Article信州大学理学部紀要 6(1): 35-36(1971)departmental bulletin pape

Notes on the Radical of a Finite Dimensional Algebra

Article信州大学理学部紀要 14(2): 95-97(1980)departmental bulletin pape

On Sexauer-Warnock Theorem

Article信州大学理学部紀要 5(1): 33-34(1970)departmental bulletin pape