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.
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 $\mathbb F_q(x)$.
It may be that since the word "field" has other uses (e.g., "vector field"), the authors wanted to emphasize that their current use was in this abstract algebra sense.