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Finite Extension


An extension field F subset= 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).

A ring extension R subset= S is called finite if S is finitely generated as a module over R. An example is the ring of Gaussian integers Z[i], which is generated by 1,i as a module over Z. The polynomial ring Z[x], however, is not a finite ring extension of Z, since all systems of generators of Z[x] as a Z-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,....

A finite ring extension is always integral.


This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Finite Extension." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/FiniteExtension.html

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