A NOTE ON GIRSTMAIR’S IRREDUCIBILITY CRITERION

Girstmair [‘On an irreducibility criterion of M. Ram Murty’, Amer. Math. Monthly112(3) (2005), 269–270] gave a generalisation of Ram Murty’s irreducibility criterion. We further generalise these criteria.

On the Irreducible Factors of a Polynomial and Applications to Extensions of Absolute Values

Polynomial factorization over a field is very useful in algebraic number theory, in extensions of valuations, etc. For valued field extensions, the determination of irreducible polynomials was the focus of interest of many authors. In 1850, Eisenstein gave one of the most popular criterion to decide on irreducibility of a polynomial over Q. A criterion which was generalized in 1906 by Dumas. In 2008, R. Brown gave what is known to be the most general version of Eisenstein-Schönemann irreducibility criterion. Thanks to MacLane theory, key polynomials play a key role to extend absolute values. In this chapter, we give a sufficient condition on any monic plynomial to be a key polynomial of an absolute value, an irreducibly criterion will be given, and for any simple algebraic extension L=Kα, we give a method to describe all absolute values of L extending ∣∣, where K is a discrete rank one valued field.

Note on Irreducible Polynomials over Finite Field

In this note we extend an irreducibility criterion of polynomial over finite fields. Weprove the irreducibility of the polynomial P(Y ) = Yn + λn−1Y n−1 + λn−2Y n−2 + · · · + λ1Y + λ0, such that λ0 6= 0, deg λn−2 = 2 deg λn−1 + l deg λi, for all i 6= n − 2 and odd integer l.

Irreducibility criterion and 2-pisot series in positive character

Let Fq((X?1)) be the field of formal power series over a finite field Fq. We characterize a pair of roots that lies outside the unit disc while all remaining conjugates have a modulus strictly less than 1. In particular, we provide a sufficient condition for a pair of formal power series to be a 2-Pisot series. We also give an irreducibility criterion over Fq [X].