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Field extensions can be generalized to Ring Extension which consists of a Ring and one of its Subring s. DEFINITIONS Given two fields ''K'' and ''L'', if ''K'' is a Subset of ''L'' and the field operations of addition and multiplication in ''K'' are the same as those in ''L'', we say that ''K'' is a subfield of ''L'', ''L'' is an '''extension field''' of ''K'' and that ''L''/''K'', read as "''L'' over ''K''", is a '''field extension'''. If ''L'' is an extension of ''F'' which is in turn an extension of ''K'', then we say ''F'' is an intermediate field or '''subextension''' of the field extension ''L''/''K''. Given a field extension ''L''/''K'' and a subset ''S'' of ''L'', we denote by ''K''(''S'') the smallest subfield of ''L'' which contains ''K'' and ''S''. We say ''K''(''S'') is generated by the Adjunction of elements of ''S'' to ''K''. If ''S'' consists of only one element ''s'' we often write ''K''(''s'') instead of ''K''({''s''}). A field extension of the form ''L''=''K''(''s'') is called a Simple Extension and ''s'' is called a Primitive Element of the extension. Given a field extension ''L''/''K'', then ''L'' can also be considered as a Vector Space over ''K''. The elements of ''L'' are the "vectors" and the elements of ''K'' are the "scalars". We add the vectors just like we add elements in ''L'', and scalar multiplication is multiplication of elements from ''L'' by elements from ''K''. The Dimension of this vector space is called the degree Of The Extension , and is denoted by {Link without Title} . An extension of degree 1 (that is, one where ''L'' is equal to ''K'') is called a trivial extension. Extensions of degree 2 and 3 are called '''quadratic extensions''' and '''cubic extensions''' respectively. Depending on whether the degree is finite or infinite the extension is called a '''finite extension''' or '''infinite extension'''. NOTES The notation ''L''/''K'' is purely formal and does not imply the formation of a Quotient Ring or Quotient Group or any other kind of division. It is often desirable to talk about field extensions in situations where the small field is not actually contained in the larger one, but is naturally embedded. For this purpose, one abstractly defines a field extension as an Injective Ring Homomorphism between two fields. ''Every'' ring homomorphism between fields is injective, so field extensions are precisely the Morphism s in the Category of fields. In the sequel, we will suppress the injective homomorphism and assume that we are dealing with actual subfields. EXAMPLES The field of ), so this extension is infinite. |
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