
290 Transcendental Numbers
give rise to exponents α
s
+ α
t
. Taken over all pairs s,t we get exponents of the form
α
1
+ α
2
,...,α
n−1
+ α
n
. The elementary symmetric polynomials of these are sym-
metric in α
1
,...,α
n
, so by Theorem 18.10 they can be expressed as polynomials in
the elementary symmetric polynomials of α
1
,...,α
n
. These in turn are expressible
in terms of the coefficients of the polynomial θ
1
whose zeros are α
1
,...,α
n
. Hence
the pairs α
s
+ α
t
satisfy a polynomial equation θ
2
(x) = 0 where θ
2
has rational co-
efficients. Similarly the sums of k of the α’s are zeros of a polynomial θ
k
(x) over Q.
Then
θ
1
(x)θ
2
(x)...θ
n
(x)
is a polynomial over Q whose zeros are ...