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Zbl 0733.11039
Llorente, P.; Nart, E.; Vila, N.
Decomposition of primes in number fields defined by trinomials. (English)
[J] Sémin. Théor. Nombres Bordx., Sér. II 3, No. 1, 27-41 (1991). ISSN 0989-5558

Let K be the number field defined by the trinomial $f(x)=x\sp n+Ax+B$ 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 $\vert (n-1,\nu\sb p(A))$ and $0<\nu\sb p(A)<\nu\sb p(B)$ where $\nu\sb p(k)$ is the exact power of p dividing the integer K. For p $\vert n$, complete results are given except when $0<\nu\sb p(B)\le \nu\sb p(A)$ and $\nu\sb p(B)\equiv 0(mod p)$ or p $\vert A$ and p $\nmid B$. The main purpose of this article is to obtain complete results when $n=p\sp m$, p $\vert A$ and p $\nmid B$. The proofs use techniques of Newton's polygon which were developed by {\it Ö. Ore} [Math. Ann. 99, 84-117 (1928; JFM 54.0191.02)]. The results also effectively determine the discriminant of K. \par Since it may be assumed that $\nu\sb p(A)\ge n-1$ and $\nu\sb p(B)\ge n$ 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.
[C.Parry (Blacksburg)]

MSC 2000:: *11R27 Units and factorization
11R09 Polynomials over global fields

Keywords: discriminant; trinomial; prime ideal decomposition; Newton's polygon

Citations: JFM 54.0191.02

Cited in: Zbl 1065.11088

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