extension field

plural extension fields

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

   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).

   1992, James G. Oxley, “Matroid Theory”, in Paperback, Oxford University Press, published 2006, page 215

   2 more examples

   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.

   1995, Terence Jackson, From Polynomials to Sums of Squares, Taylor & Francis, page 56

   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.

   1998, Neal Koblitz, Algebraic Aspects of Cryptography, Volume 3, Springer, page 53

   Synonyms: extension