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!

  • $\begingroup$ I would consider $ex-\pi$ to be an algebraic, in fact, linear function that just happens to have transcendental numbers for coefficients... $\endgroup$ – J. M. is a poor mathematician Jan 13 '12 at 1:27

There is no difference between allowing rational functions to requiring polynomial functions as coefficients. To see this consider $y$ a root of some polynomial with rational coefficients, say $f(y)=0$. Let $h(x)$ be the product of all of the denominators of the coefficients. Then $y$ is a root of $g(y)=h(x)f(y)$ whose coefficients are polynomials (we've just cleared denominators).

The same argument shows that the set of all $z \in \mathbb{C}$ such that $f(z)=0$ is the same whether $f(x)$ ranges over $\mathbb{Z}[x]$ or $\mathbb{Q}[x]$.

As for whether you must use rational coefficients...I'm not sure what "most" mathematicians mean when they speak of an algebraic function. I'm apt to go with J.M. and allow coefficients to be any complex number. Technically these functions are transcendental over $\mathbb{C}[x]$. However, Wikipedia is correct. If no base is specified, "Transcendental function" means "transcendental over $\mathbb{Q}[x]$ (or equivalently $\mathbb{Q}(x)$).


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