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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.

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

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