# What is an algebraic field?

I am studying programming form the book "Introduction to algorithms". There was said in chapter "Polynomials and the FFT" that

A polynomial in the variable $x$ over an algebraic field $F$ is a representation of a function $A(x)$ as a formal sum: $$A(x)=\sum_{j=0}^{n-1}a_jx^j.$$

But what is the definition of an algebraic field? I have heard about algebraically closed fields and algebraic number fields but never heard about algebraic fields.

• I think they just mean "a field as we mean in algebra, as opposed to a vector field or force field or anything else". – Arthur Aug 4 '17 at 17:47

In contemporary usage, "algebraic field" does not have any precise meaning. As you say, "a field $F$ algebraic over a field $E$" does have a precise meaning, namely, that every element $x\in F$ is algebraic over the field $E$. Note that $F$ need not be of finite degree over $E$. Yes, an "(algebraic) number field" is of finite degree over $\mathbb Q$. A "global field" is either a number field or a "function field", the latter being a finite extension of $\mathbb F_q(x)$.