NEWTON POLYGONS AND p-INTEGRAL BASES OF QUARTIC NUMBER FIELDS

Abstract

In this paper, based on techniques of Newton polygons, a result which allows the computation of a p integral basis of every quartic number field is given. For each prime integer p, this result allows to compute a p-integral basis of a quartic number field K defined by an irreducible polynomial P (X) = X 4 + aX + b ∈ Z[X] in methodical and complete generality.

Two families of monogenic $S_4$ quartic number fields

“…Let t be the x-coordinate of the initial vertex of S. We define the residual polynomial attached to S to beR λ (f )(y) = c t + c t+e y + · · · + c t+(d−1)e y d−1 + c t+de y d ∈ F p [x]/φ(x)[y].Now we state the Theorem of the index, our key tool in proving monogeneity. This is Theorem 1.9 of[6].…”

“…In order to have ind x+1 (f a,b ) = 0, we need v 3 (b − a + 1) = 1. This is satisfied by the following pairs (a, b) in Z/9Z × Z/9Z: (1,3), (1,6), (4, 0), (4, 6), (7, 0), (7,3). The residual polynomial is linear and hence separable.…”

“…We prove monogeneity with a simple application of the Montes algorithm. We follow [6] for our exposition of the algorithm. Those interested in more general situations are advised to consult [21].…”

“…The principal x-polygon merely excludes the side between (4, 0) and (6,0). Further, accounting for (1, 1), (2, 1), and (1, 2), we see ind…”

Discriminants of Chebyshev radical extensions

“…Their method employs a more refined variation of the Newton polygon, called the φ-Newton polygon, which captures arithmetic data attached to each irreducible factor φ of Φ. In this section we outline their methods and terminology following the presentation of El Fadil, Montes, and Nart [5]. Notation 1.…”