364 APPENDIX A. CLASSIFICATION OF REAL NUMBERS

where the pi are elementary symmetric polynomials defined as the coefficients of p̂(x) =∏

nj=1 (x− x j) with pk (x1, ...,xn) of degree k since it is the coefficient of xn−k. Earlier we

had them ± these coefficients. Thus

f (anα1, · · · ,anαn) = ∑k1+···+kn=d

ak1···kn pk11 (anα1, · · · ,anαn) · · · pkn

n (anα1, · · · ,anαn)

Now the given polynomial in ∗, p(x) is of the form

an

n

∏j=1

(x−α j)≡ an

(n

∑k=0

pk (α1, · · · ,αn)xn−k

)

= anxn +an−1xn−1 + · · ·+a1x+a0

Thus, equating coefficients, an pk (α1, · · · ,αn) = an−k. Multiply both sides by ak−1n . Thus

pk (anα1, · · · ,anαn) = ak−1n an−k

an integer. Therefore,

f (anα1, · · · ,anαn) = ∑k1+···+kn=d

ak1···kn pk11 (anα1, · · · ,anαn) · · · pkn

n (anα1, · · · ,anαn)

and each pk (anα1, · · · ,anαn) is an integer. Thus f (anα1, · · · ,anαn) is indeed an integer.From this, it is obvious that f (α1, · · · ,αn) is rational. Indeed, from 1.3,

f (α1, · · · ,αn) = ∑k1+···+kn=d

ak1···kn pk11 (α1, · · · ,αn) · · · pkn

n (α1, · · · ,αn)

Now multiply both sides by aMn , an integer where M is chosen large enough that

aMn f (α1, · · · ,αn)

= ∑k1+···+kn=d

ah(k1,...,kn)n ak1···kn pk1

1 (anα1, · · · ,anαn) · · · pknn (anα1, · · · ,anαn)

where h(k1, ...,kn) is some nonnegative integer. Then the right side is an integer. Thusf (α1, · · · ,αn) is rational. If the f had rational coefficients, then m f would have inte-ger coefficients for a suitable m and so m f (α1, · · · ,αn) would be rational which yieldsf (α1, · · · ,αn) is rational.

A.3 Transcendental NumbersMost numbers are like this, transcendental. Here the algebraic numbers are those whichare roots of a polynomial equation having rational numbers as coefficients, equivalentlyinteger coefficients. By the fundamental theorem of algebra, all these numbers are in Cand they constitute a countable collection of numbers in C. Therefore, most numbers in Care transcendental. Nevertheless, it is very hard to prove that a particular number is tran-scendental. Probably the most famous theorem about this is the Lindermannn Weierstrasstheorem, 1884.

Theorem A.3.1 Let the α i be distinct nonzero algebraic numbers and let the ai benonzero algebraic numbers. Then ∑

ni=1 aieα i ̸= 0.