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Resultants of cyclotomic polynomials. (English) Zbl 0188.34002

The author makes use of the lemma quoted in his previous review [Proc. Am. Math. Soc. 24, 482–485 (1970; Zbl 0188.34001)] to prove the following theorem concerning the resultant \(\rho(F_m,F_n)\) of two cyclotomic polynomials. If \(m>n>1\) and \((m,n)>1\), then \(\rho(F_m,F_n)=p^{\varphi(n)}\) if \(m/n\) is a power of a prime \(p\), and \(\rho(F_m,F_n)=1\) otherwise. Also, if \(m>n>1\) and \((m,n)=1\), then \(\rho(F_m,F_n)=1\). Here \(F_n(x)\) is the primary polynomial whose roots are the primitive \(n\)-th roots of unity.

MSC:

11C08 Polynomials in number theory

Citations:

Zbl 0188.34001
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References:

[1] Tom M. Apostol, Euler’s \?-function and separable Gauss sums, Proc. Amer. Math. Soc. 24 (1970), 482 – 485. · Zbl 0188.34001
[2] L. E. Dickson, H. H. Mitchell, H. S. Vandiver and G. E. Wahlin, Algebraic numbers, Bulletin of the National Research Council, vol. 5, part 3, no. 28, National Academy of Sciences, 1923. · JFM 49.0109.01
[3] Fritz-Erdmann Diederichsen, Über die Ausreduktion ganzzahliger Gruppendarstellungen bei arithmetischer Äquivalenz, Abh. Math. Sem. Hansischen Univ. 13 (1940), 357 – 412 (German). · Zbl 0023.01302
[4] Hans Rademacher, Lectures on elementary number theory, A Blaisdell Book in the Pure and Applied Sciences, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964. · Zbl 0119.27803
[5] B. L. van der Waerden, Moderne algebra. Vol. I, 2nd rev. ed., Springer, Berlin, 1937; English transl., Ungar, New York, 1949. MR 2, 120; MR 10, 587. · JFM 63.0082.06
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