Suppose F is a subfield of the field K. Then K is called an extension field of F. So, for instance, GF(4) and GF(8) are extension fields of GF(2), although GF(8) is not an extension field of GF(4).

This extension field of F always contains a root of f in the sense that if K=F[x]/(f(x)) then x is a root of f(y) in K[y]. It then follows that any polynomial will have roots, either in the original field of its coefficients or in some extension field.

An extension field, by which we mean a bigger field containing F, is automatically a vector space over F. We call it a finite extension if it is a finite vector space. By the degree of a finite extension we mean its dimension as a vector space. One common way of obtaining extension fields is to adjoin an element to F: we say that K=F(α) if K is the field consisting of all rational expressions formed using α and elements of F.

Extension field

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