Meaning of extension field | Babel Free

Definitions

A field L which contains a subfield K, called the base field, from which it is generated by adjoining extra elements.

Examples

“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#61;F#91;x#93;#47;(f(x)) then x is a root of f(y) in K#91;y#93;. 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#61;F(#92;alpha) if K is the field consisting of all rational expressions formed using #92;alpha and elements of F.”

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