4 1. SYMMETRI C FUNCTION S
i.e., tha t
n
(4.2) £ ( - l ) r e
r
/ i
n
_
r
= 0
for eac h n 1. Sinc e th e e
r
ar e algebraicall y independen t w e ma y defin e a rin g
homomorphism u : A A by
u;(er) = h
r
(r 1)
and th e symmetr y o f th e relation s (4.2 ) show s tha t u)(h r) = e
r
fo r eac h r , s o tha t
LU2
= 1, i.e., a; is an involution . Fro m (3.2 ) i t follow s tha t
(4.3) A = Z[/ii , /12,... ], and fti, h
2
,... ar e algebraically independent over Z .
Equivalently, th e product s
ft A = h
Xl
h\2 " =v(e\)
form a third Z-basi s o f A.
5. Powe r sum s
For eac h r 1, the rt h power sum i s
pr = m
{r)
=^2,x\.
Their generatin g functio n i s
(5.i)
^ ) =
E^r"1
=
E i ^ '
r l i
i.e.,
(5.2) P( ) = H'(t)/H(t)
by (4.1), where if'(t ) i s the derivativ e o f H(t) wit h respec t t o t. I f we write this i n
the for m
H'(t) = P(t)H(t)
and pic k ou t th e coefficien t o f t
n~l
o n eithe r side , we shall obtai n
n
(5.3) nh
n
= y^p
r
ft
n
_
r
r= l
for eac h n 1. I t follow s fro m (5.3) , b y inductio n o n n , tha t h
n
£ Q[pi,.. . ,p
n
]
and p
n
G Z[fti,..., h n]. Henc e
(5.4) A Q = Q[pi,P2 ]
a
^ Pi?P2 , &r e algebraically independent over Q.
Equivalently, th e power-su m product s
Px = P\!P\2 •' '
form a Q-basi s o f AQ . However , the y d o not for m a Z-basi s o f A : fo r example ,
e2 = \{p\ P2) does no t hav e integra l coefficient s a s a linea r combinatio n o f th e
Px-
Likewise, fro m (5.1) an d (3.1) w e hav e
(5.5) P(-t) = E'(t)/E(t).
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