I have some minor questions about definition of algebraic and transcendental functions:
An algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable $x$ is a solution $y$ for an equation $ a_n(x)y^n+a_{n-1}(x)y^{n-1}+\cdots+a_0(x)=0 $ where the coefficients $a_i(x)$ are polynomial functions of $x$ with rational coefficients. A function which is not algebraic is called a transcendental function.
My understanding of an algebraic function is that it is defined to be an element of the algebraic closure of the field of rational functions, i.e. it is the root to a polynomial with coefficients being rational functions. Is it true that my understanding is equivalent to Wiki's version that an algebraic function is the root to a polynomial with coefficients being polynomial functions?
Are the coefficients of $a_i$ by default rational numbers? Do people use real number and complex numbers often?
Thanks and regards!