IRRATIONAL RESOLVENTS.
Up to the present we have considered the effect on the Galoisian Group of an equation, of adjoining roots of rational resolvents to the cur rent domain of rationality. In many investiga tions it is important to consider the adjunction of roots of equations which may not be rational functions of the roots of the given equation. Equations whose roots are not rational functions of the roots of the given equation are called irrational resolvents when used in the solution of the given equation. A theorem which lies at the foundation of this subject is due to Kronecker. Let f(x)=0, g(y)=0 be two ra tional irreducible equations for the domain R of degrees m, n respectively. If on adjoining a root xi, of f g(y) becomes reducible, the adjunc tion of a root of g=0, will make f(x) re ducible. If cp(x), 4'(x) of degrees a, re spectively, be the irreducible factors for the new domains that xi yi satisfy, then m n a p* As an important corollary of Kronecker's theorem we have: Let the adjunction of yi reduce the group G of f(x) =0 to an invariant subgroup of index i. Then n is a multiple of i and hence never less than i. When n=i (and this is always the case if n is a prime) g (y) =0 is a rational resolvent.
Another theorem of great im importance in this connection is due to Jordan. I the adjunction of all the roots of g(y)== 0 G to a sub group of index s, the adjunction of all the roots of f(x) reduces the group H of g(y) to a subgroup H, of index k. The two groups Gi, Hs are invariant and i= k. Finally, when H is simple g(y)= 0 is a rational resolvent.
Application to Some Celebrated Problems. — The Delian Problem or duplication of the cube requires the solution of x'-2) by rule and compass. The construction of the regular polygons by rule and compass is an other famous problem of antiquity. Its solu tion depends upon the irreducible equation of degree elt(n) already referred to. That the
Delian Problem is impossible follows at once from the theorem: In order that a root, real or imaginary, of an irreducible equation f(x) = 0 can be constructed geometrically it is necessary that the degree of f be a power of two. From this theorem we also conclude: The necessary and sufficient condition that a regular polygon of n sides can be constructed by rule and compass is that the totient of a is a power of two.
Another famous question is the Casus Irre ducibilis of cubic equations. The theory of irrational resolvents enables us to prove readily the following general theorem: An irreducible equation of degree a whose roots are all real can never be solved by real radicals alone if n contains other factors than two.
That the casus irreducibilis is indeed such follows as corollary of this theorem.
Holder's Theorem.— One of the most im portant and fundamental contributions to Ga lois' theory in recent years is a theorem of Holder. Speaking roughly, it asserts that how ever the solution of a given equation f(x) =0 be conducted, sometime in the course of the solution certain simple equations whose groups are uniquely determined and known in advance must be employed. When the group of t(x)— 0 is simple (in which case we say f(x) is simple) it can be solved by no other simple equation g 0 essentially different from 1= 0. The solution of any given equation therefore depends upon a chain of simple equa tions. But of all simple equations belonging to a given group certain ones will enjoy peculiar properties which will recommend their selection as normal equations. The reduction of the given equation to these normal equations is a problem by itself.