WHEN CAN THE PRIMITIVE ELEMENT BE WRITTEN AS A SUM OF TWO ALGEBRAIC ELEMENTS ADJOINED TO THE FIELD OF RATIONAL NUMBERS?
Mohamad Medhat Moussa · scholarworks - UTEP (The University of Texas at El Paso) · 2015
Given a field F and elements \\alpha and \\beta not in F, then F(\\alpha, \\beta) is the smallest field containing \\alpha,\\beta, and F. A simple extension is a field extension which is generated by the adjunction of a single element. The Primitive Element Theorem says that if F is a field of characteristic 0, and \\alpha and \\beta are algebraic over F, then there is an element \\gamma in F(\\alpha ,\\beta ) such that F(\\alpha ; \\beta ) = F(\\gamma). When can we say that \\gamma=\\alpha+\\beta? We will introduce some situations where \\gamma=\\alpha+\\beta is true and some when this is not true, where F is the field of rational numbers Q.