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Decomposition of primes in number fields defined by trinomials. (English) Zbl 0733.11039

Let K be the number field defined by the trinomial \(f(x)=x^ n+Ax+B\) where A and B are integers. The objective of the article is to determine the prime ideal decomposition of a rational prime p in K. For p \(\nmid n\), complete results are given except when p \(| (n-1,\nu_ p(A))\) and \(0<\nu_ p(A)<\nu_ p(B)\) where \(\nu_ p(k)\) is the exact power of p dividing the integer K. For p \(| n\), complete results are given except when \(0<\nu_ p(B)\leq \nu_ p(A)\) and \(\nu_ p(B)\equiv 0(mod p)\) or p \(| A\) and p \(\nmid B\). The main purpose of this article is to obtain complete results when \(n=p^ m\), p \(| A\) and p \(\nmid B\). The proofs use techniques of Newton’s polygon which were developed by Ö. Ore [Math. Ann. 99, 84-117 (1928; JFM 54.0191.02)]. The results also effectively determine the discriminant of K.
Since it may be assumed that \(\nu_ p(A)\geq n-1\) and \(\nu_ p(B)\geq n\) are not simultaneously satisfied, no exceptions occur for \(n=3\). For \(n=4\) and 5, results are also obtained for the exceptional cases mentioned above.

MSC:

11R27 Units and factorization
11R09 Polynomials (irreducibility, etc.)

Citations:

JFM 54.0191.02
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References:

[1] Llorente, P. - Nart, E., Effective determination of the decomposition of the rational primes in a cubic field, Proc. Amer. Math. Soc.87 (1983), 579-585. · Zbl 0514.12003
[2] Llorente, P. - Nart, E. - Vila, N., Discriminants of number fields defined by trinomials, Acta Arith.43 (1984), 367-373. · Zbl 0493.12010
[3] Ore, Ö., Zur Theorie der algebraischen Körper, Acta Math.44 (1923), 219-314. · JFM 49.0698.04
[4] Ore, Ö., Newtonsche Polygone in der Theorie des algebraischen Körper, math. Ann.99 (1928), 84-117. · JFM 54.0191.02
[5] Swan, R.G., Factorization of polynomials over finite fields, Pacific J. Math.12 (1962), 1099-1106. · Zbl 0113.01701
[6] Vélez, W.Y., The factorization of p in Q(a1/pk ) and the genus field of Q(a1/n), Tokyo J. Math.11 (1988), 1-19. · Zbl 0664.12003
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