Value of algebraic functions at algebraic numbers

Question

I have read that the value of an algebraic function at an algebraic number (root of a polynomial with integer coefficients) is also an algebraic number, and that all polynomials are algebraic functions. However, if I take the polynomial e*x, its value at algebraic numbers is not another algebraic number.

Can anyone tell me what I'm missunderstanding?

Thank you.

Answer 1

I'm going to assume that this is the definition of an algebraic function (following this wiki article):

Definition: A function $f(x)$ is algebraic if it is continuous in its domain and $$a_n(x)f(x)^n+\cdots +a_1(x)f(x)+a_0(x)=0$$ where each $a_i(x)$ is a polynomial with integer coefficients.

(Although I don't think that the continuity assumption matters much). The "integer coefficients" assumption is crutial. It is an analogy to the usual algebraic number definition. Now the statement

If $f(x)$ is an algebraic function and $a$ an algebraic number then $f(a)$ is algebraic number as well.

is true in this setup. It is not true if we omit "integer coefficients" assumption. And the statement

All polynomials are algebraic

is false as your $f(x)=ex$ example shows. The statement should say

All rational polynomials are algebraic

This one is quite easy to prove. Also I think that

All polynomials with algebraic coefficients are algebraic

is true but I couldn't figure out the details. And in fact I think that for polynomials this is "if and only if" meaning if a polynomial is algebraic then its coefficients are algebraic.