Solution to a Problem of Lubelski and an Improvement of a Theorem of His

• Autorzy: Schinzel A.

Identyfikatory

DOI

10.4064/ba59-2-2

• Języki publikacji: EN

Abstrakty

EN

The paper consists of two parts, both related to problems of Lubelski, but unrelated otherwise. Theorem 1 enumerates for a=1,2 the finitely many positive integers D such that every odd positive integer L that divides x2+Dy2 for (x,y)=1 has the property that either L or 2aL is properly represented by x2+Dy2. Theorem 2 asserts the following property of finite extensions k of Q: if a polynomial f∈k[x] for almost all prime ideals p of k has modulo p at least v linear factors, counting multiplicities, then either f is divisible by a product of v+1 factors from k[x]∖k, or f is a product of v linear factors from k[x].

Słowa kluczowe

EN

binary quadratic form; polynomial modulo a prime ideal

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2011
• Tom: Vol. 59, no 2
• Strony: 115--119
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Schinzel A.

• Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland

Bibliografia

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• [7] —, Zur Reduzibilität von Polynomen in der Kongruenztheorie, Acta Arith. 1 (1936), 169–183.
• [8] J. Oesterlé, Nombres de classes des corps quadratiqes imaginaires, Astérisque 121–122 (1985), 309–323.
• [9] H. M. Stark, On complex quadratic fields with class number two, Math. Comp. 29 (1975), 289–302.
• [10] H. Weber, Lehrbuch der Algebra, Bd. III, reprint Chelsea, 1961.