Finite Extension (original) (raw)

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An extension field F subset= K is called finite if the dimension of as a vector space over (the so-called degree of over ) 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 of complex numbers is a finite extension, of degree 2, of the field 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 ).

A ring extension R subset= S is called finite if is finitely generated as a module over . An example is the ring of Gaussian integers Z[i] , which is generated by 1,i as a module over . The polynomial ring Z[x] , however, is not a finite ring extension of , since all systems of generators of Z[x] as a -module have infinitely many elements: in fact they must be composed of polynomials of all possible degrees. The simplest generating set is the sequence 1,x,x^2,....