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In this paper, for a henselian valued field (K , v) of arbitrary rank and an extension w of v to K (X) , we use abstract key polynomials for w to obtain distinguished pairs and saturated distinguished chains. [ABSTRACT FROM AUTHOR]

In this paper, for a valued field (K , v) of arbitrary rank and an extension w of v to K (X) , a relation between induced complete sequences of abstract key polynomials and MacLane-Vaquié chains is given. [ABSTRACT FROM AUTHOR]

A well known result of Schur states that if n is a positive integer and a0, a1,...,an are arbitrary integers with a0an coprime to n!, then the polynomial is irreducible over the field ℚ of rational numbers. In case each ai = 1, it is known that the Galois group of fn(x) over ℚ contains An, the alternating group on n letters. In this paper, we extend this result to a larger class of polynomials fn(x) which leads to the construction of trinomials of degree n for each n with Galois group Sn, the symmetric group on n letters. [ABSTRACT FROM AUTHOR]

A well known result of Schur states that if n is a positive integer and a0, a1,...,an are arbitrary integers with a0an coprime to n!, then the polynomial is irreducible over the field ℚ of rational numbers. In case each ai = 1, it is known that the Galois group of fn(x) over ℚ contains An, the alternating group on n letters. In this paper, we extend this result to a larger class of polynomials fn(x) which leads to the construction of trinomials of degree n for each n with Galois group Sn, the symmetric group on n letters. [ABSTRACT FROM AUTHOR]