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.

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The methods of classical invariant theory are used to construct generic polynomials for groups S 5 and A 5 , along with explicit reductions to specializations of the generic polynomials defining any desired field extension with those groups. The classical invariant theory of binary formsThe classical invariant theory of binary forms explores the invariants under the action of SL 2 (K) in a field K of characteristic 0, but we will extend that to consideration of characteristics other than 2, 3, or 5. Initially, however, it will do no harm to think of K as Q or a field of rational functionsIf a binary form of homogenous degree n in x and y has a nonzero coefficient for x n , then by setting y to 1, we obtain a polynomial in x of degree n. Substituting x/y for x and multiplying by y n gives the form back again. Hence the invariant theory of binary forms is also an invariant theory for polynomials in one variable.Classical invariant theory actually explores a larger class of invariants, which include the covariants. A covariant of a binary form f is a polynomial of homogenous degree r (called the order) in x and y, with coefficients which are of degree s in the coefficients of f , which is invariant in a particular sense under SL 2 (K). The invariants themselves are then the covariants order 0.If M = ( a b c d ) is a matrix with determinant 1, and g a form, then M(g) is the form obtained by substituting x → ax + by, y → cx + dy for x and y. Let C(f ) be a function from forms of order n to forms of order r, whose coefficients are defined by means of homogenous polynomial functions of

Formulas for calculating the Riesz function, introduced by Marcel Riesz in connection with the Riemann hypothesis, are derived; and the behavior of the Riesz function is discussed.