D. The General Equation of Degree n.
If F is a field, the collection of rational expressions in the variables ut, u2, . . . , un with coefficients in F is a field F(u1? u2, . . . , By the general equation of degree n we mean the equation
Let E be the splitting field of fl(x) over are the roots of f(x) in E, then
We shall prove that the group of E over F (Uj, u2, . . . , un ) is the symmetric group.
De the elementary symmetric functions, i.e., (x-x ,)(x-x . . (x-x,) = polynomial in al, . . . ,a,, then g(a1 , a2, . . . ,a,) = 0 only if g is the
Let F( x x , . , . , x ) be the field generated from F by the zero polynomial. For if then this relation would hold also if the x. were replaced by the v. . Thus, romwhichitfollows that g is identically zero.
Between the subfield and we set up the following correspondence: Let
We make this correspond to ping of be an element of This is clearly a map Moreover, if
But this implies by the above that
f(Ui,- • • ,un)/g(u1Ju,J.. . ,uB)
so that It follows readily from this that themappingof is an isomorphism. But under this correspondence f(x) corresponds to f * ( x). Since El and F(Xj, x2, xn) are respectively splitting fields of f(x) and f * (x), by Theorem 10 the isomorphism can be extended to an isomorphism between E and F
Therefore, the group of E s isomorphic to the group of F over Each permutation of which and, therefore, induces an automorphism of fixed. Conversely, each automorphism of roots which leaves fixed must permute the and is completely determined by the
Thus, the group of permutation it effects on over is the symmetric group on n letters. Because of and E, the group for E over the isomorphism between symmetric group for n > 4 is not solvable, we obtain from the theorem on solvability of equations the famous theorem of Abel:
THEOREM 6. The group of the general equation of degree n is the symmetric group on n letters. The general equation of degree n is not solvable by radicals if n > 4.