Various Basic Theorems

Abstract

Let A be a vector space over F that has a law of multiplication which, together with the existing law of addition, makes A an associative ring. We call A an algebra over F, or simply an F -algebra, if $\,c({\alpha}{\beta}) =(c{\alpha}){\beta}={\alpha}(c{\beta})\,$ for every $\,c\in F$ and $\,{\alpha},\,{\beta}\in A.$ If A has an identity element $1_A,$ then identifying $\,c\,$ with $\,c1_A,$ we can view F as a subfield of A. Notice that $\,c{\alpha}=(c1_A){\alpha}={\alpha}(c1_A) ,$ and so two laws of multiplication for the elements of F (one in the vector space and the other in the ring) are the same. Every field extension of F can naturally be viewed as an F-algebra.