Regular extension

In <a href="/facts/Field_theory_(mathematics)/xAjAS4ko">field theory</a>, a branch of algebra, a <a href="/facts/Field_extension/L7yIDtyK">field extension</a> 
 
 
 
 L
 
 /
 
 k
 
 
 {\displaystyle L/k}
 
 is said to be regular if k is <a href="/facts/Algebraically_closed/HuVQClok">algebraically closed</a> in L (i.e., 
 
 
 
 k
 =
 
 
 
 k
 ^
 
 
 
 
 
 {\displaystyle k={\hat {k}}}
 
 where 
 
 
 
 
 
 
 k
 ^
 
 
 
 
 
 {\displaystyle {\hat {k}}}
 
 is the set of elements in L algebraic over k) and L is <a href="/facts/Separable_extension/9KBvXmz8">separable</a> over k, or equivalently, 
 
 
 
 L
 
 ⊗
 
 k
 
 
 
 
 k
 ¯
 
 
 
 
 {\displaystyle L\otimes _{k}{\overline {k}}}
 
 is an integral domain when 
 
 
 
 
 
 k
 ¯
 
 
 
 
 {\displaystyle {\overline {k}}}
 
 is the algebraic closure of 
 
 
 
 k
 
 
 {\displaystyle k}
 
 (that is, to say, 
 
 
 
 L
 ,
 
 
 k
 ¯
 
 
 
 
 {\displaystyle L,{\overline {k}}}
 
 are <a href="/facts/Linearly_disjoint/Totpn0fa">linearly disjoint</a> over k).

Regular extension open-in-new

Regular extension