A Game with Two Players Choosing the Coefficients of a Polynomial.

We consider some versions of a game when two players Nora and Wanda in some order are choosing the coefficients of a degree d polynomial. The aim of Nora is to get a polynomial which has no roots in some field or, more generally, is irreducible over that field or, even more generally, has the largest possible Galois group S d , while the aim Wanda is the opposite. We show that in order to obtain an irreducible polynomial for Nora it suffices to have the last move. However, to ensure that the splitting field of the resulting polynomial with integer coefficients has Galois group S d Nora needs to have at least three moves for each even d ≥ 4 . For d = 4 we show that Nora can always get the Galois group S 4 if Nora starts and they play alternately. [ABSTRACT FROM AUTHOR]