M.Sc Student | Rabayev Daniel |
---|---|

Subject | Polynomials with Roots Mod n for All Positive Integers n |

Department | Department of Mathematics |

Supervisor | Professor Jack Sonn |

Full Thesis text |

We are interested in polynomials with
integer coefficients that have roots mod n for all n, or equivalently, have
roots in the *p*-adic numbers for all p. We consider only nontrivial
polynomials, i.e., having no rational roots. It is known that such a polynomial
cannot be irreducible, hence must be a product of two or more irreducible
factors. Furthermore, the Galois group of such a polynomial is n-coverable,
i.e., a union of n conjugacy classes of n proper subgroups, where n is the
number of the irreducible factors of the polynomial. Such polynomials
consisting of two irreducible factors are of special interest. In that case, it
can be shown that among the symmetric and alternating groups, only *S*_{n},
where 2<*n*<7, and *A*_{n} where 3<n<9 can occur
as Galois groups of such polynomials. We have found explicit polynomials for
these groups except for *A*_{7} and *A*_{8}. In
general, a sufficient condition that a Galois extension k of the Rational
numbers is the splitting field of such a polynomial is that all the
decomposition groups are cyclic. This sufficient condition was used in all of
the above cases except for *A*_{6}, where a specific 2-covering
may be used. In this work we will prove (by constructing explicit polynomials)
that the groups *S*_{4}, *S*_{5}, *S*_{6},
*A*_{4}, *A*_{5}, *A*_{6} are
2-coverable and can be realized as Galois groups of polynomials having a root
in the *p*-adic numbers for all prime number p.