4 research outputs found
On the Common Index Divisors of a Dihedral Field of Prime Degree
A criterion for a prime to be a common index divisor of a
dihedral field of prime degree is given. This criterion is used to determine
the index of families of dihedral fields of degrees 5 and 7
Enumerative Galois theory for cubics and quartics
We show that there are monic, cubic
polynomials with integer coefficients bounded by in absolute value whose
Galois group is . We also show that the order of magnitude for
quartics is , and that the respective counts for , ,
are , , . Our work establishes
that irreducible non- cubic polynomials are less numerous than reducible
ones, and similarly in the quartic setting: these are the first two solved
cases of a 1936 conjecture made by van der Waerden
Some Polynomials over Q(t) and their Galois groups
Abstract. Examples of polynomials with Galois group over Q(t) corresponding to every transitive group through degree eight are calculated, constructively demonstrating the existence of an infinity of extensions with each Galois group over Q through degree eight. The methods used, which for the most part have not appeared in print, are briefly discussed. 1