G. Applications and Examples to Theorem 13.

Theorem 13 is very powerful as the following examples show:

1) Let k be a field and consider the field E = k (x ) of all

rational functions of the variable x. If we map each of the functions

f (x ) of E onto f(-) we obviously obtain an automorphism of E. Let us x

If o and j are two automorphisms of E, then the mapping <r( r( x)) written briefly or is an automorphism, as the reader may readily verify.

 

We shall call gt the product of <7 and r. If a is an automorphism consider the following six automorphisms where f (x 1 is macoed onto and call F the

 

We contend: F = S and (E/F ) = 6.

Indeed, from Theorem 13 we obtain (E/F) > 6. Since S C F it suffices to prove (E7 S) < 6. Now E = S(x). It is thus sufficient to find some 6-th degree equation with coefficients in S satisfied by x. The following one is obviously satisfied;

 

The reader will find the study of these fields a profitable exer­cise. At a later occasion he will be able to derive all intermediate fields.

 

2) Let k be a field and E = k(x lr x2,. .. ,xn) the field of all rational functions of n variables x , x , . . . , x^. If (^ , v2, . . . , v^ ) is a permutation of (1, 2,. . . , n) we replace in each function f (x ^ x 3> - • • , Xn)

of E the variable x. by x„ , x „ by X .. ,. ,x by x„ The mapping of E

vx i J v2          n          i/n

fixed point field. F consists of all rational functions satisfying

 

It suffices to check the first two equalities, the others being conse­quences. The function belongs to F as is readily seen. Hence, the field S = k (1) of all rational functions of 1 will belong to F.

Onto itself obtained in this way is obviously an automorphism and we may construct n ! automorphisms in this fashion (including the identity). Let F be the fixed point field, that is, the set of all so-called "symmetric functions." Theorem 13 shows that (E/F) >.n ! . Let us in­troduce the polynomial:

 

where

 

and more generally ai is ( - 1) times the sum of all products of 1 differ- erent variables of the set x,, x„, . . . , x . The functions a„ a„. . . . , a

1 2       n          " 2'       ' n

We construct to this effect the following tower of fields:

 

by the definition

 

are called the elementary symmetric functions and the field S = k ( aj, a2,. . . , a ) of all rational functions of a,, a2, . . . , is obviously a part of F. Should we suceed in proving ( S ) < n ! we would have shown S = F and (E/F) = n ! .

It would be sufficient to prove ( Si l/S, ) < i or that the generator x. for S. j out of Sj satisfies an equation of degree i with coefficients in Sj.

Such an equation is easily constructed. Put and Ffl(t) = f(t). Performing the division we see that F{ (t ) is a polynomial in t of degree i whose highest coefficient is 1 and whose coefficients are polynomials in the variables

aj, a2,. .. , a,, and x1+1, xi+2 , . . . , xn. Only integers enter as coefficients in these expressions. Now Xj is obviously a root of F, (t ) = 0.

Now let g(x1,x2J.,.,xn)bea polynomial in Xj,x2,... , xn. Since Fj ( Xj ) = 0 is of first degree in Xj , we can express Xj as a polynomial of the a. and of X2, x3< . . . , Xn . We introduce this expression B(x!» x2> • • • » xn)- ^nce F2 (x2 ) = ® we can exPress x2 or higher

powers as polynomials in x3,. . . , xn and the Sj. Since F3 ( x3) = 0 we can express x| and higher powers as polynomials of x4, x5,. • • , xn and the ar Introducing these expressions in g( x1? x2,. . . , xn) we see that we can express it as a polynomial in the x^ and the a^ such that the degree in Xj is below i. So g( x1? x2, . . . , xn) is a linear combination of the following n ! terms:

 

 

The coefficients of these terms are polynomials in the a; . Since the expressions (6) are linearly independent in S (this is our previous result), the expression is unique.

This is a generalization of the theorem of symmetric functions in its usual form. The latter says that a symmetric polynomial can be written, as a polynomial in ap a„ . . . , a„. Indeed, if g(xj, . . . , xn) is symmetric we have already an expression as linear combination of the terms (6) where only the term 1 corresponding to y = v2 = . . . = fn = 0 has a coefficient ^ 0 in S, namely, g( Xj, . . . , xn), So g( xp x2,. . . , xn) is a polynomial in a„ a2, . . . , a,.

Hut our theorem gives an expression of any polynomial, symmetric

or not.