That unnameable confusing thing about Field extensions

A first incursion into the concept of field extensions proves to be rather confusing. And I’ll tell you why. Things which seem difficult, and sometimes impossible to prove, are proved very easily using concepts of vector spaces. For example, we know that is a finite extension of . Prove that is also algebraic over .Continue reading “That unnameable confusing thing about Field extensions”

A generalization of “all members of a simple field extension of F are algebraic over F”

The following is a powerful powerful theorem: Let be a simple field extension of field , where is algebraic over . Then every is also algebraic over . Also, . The proof I think is most brilliant. I am not going to provide you with a proof here. I am just going to try andContinue reading “A generalization of “all members of a simple field extension of F are algebraic over F””