An extension field F⊆K is called finite if the dimension of K as a vector space over F (the so-called degree of K over F) is finite. A finite field extension is always algebraic. Note that "finite" is a synonym for "finite-dimensional"; it does not mean "of finite cardinality" (the field C of complex numbers is a finite extension, of degree 2, of the field R of real numbers, but is obviously an infinite set), and it is not even equivalent to "finitely generated" (a transcendental extension is never a finite extension, but it can be generated by a single element as, for example, the field of rational functions F(x) over a field F).
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