1.6. Quadratic field extensions 19 Proof. It is clear that F F ( a), so what has to be shown is that F ( a) is a field. But note that if x1,x2,y1,y2 F , then (x1 + y1 a) (x2 + y2 a) = (x1x2 + y1y2a) + (x1y2 + x2y1) a F ( a). Similar arithmetic shows that the sum and difference of any two elements of F ( a) belongs to F ( a). Also, assuming x2 + y2 a = 0, x1 + y1 a x2 + y2 a = x1 + y1 a x2 + y2 a · x2 y2 a x2 y2 a = (x1x2 y1y2a) x2 2 + y2 2 + (−x1y2 + x2y1)√ x2 2 + y2 2 a F ( a), completing the proof that F ( a) is a field. Of course if a F , then F ( a) = F . When equality fails, we will say that F ( a) is a proper extension of F . Also note that we could iterate the above process. For example, suppose a F , a 0, and a / F . So we form the proper extension F ( a). Now suppose b F ( a), b 0, and b / F ( a). Therefore we can form the proper extension F ( a)( b) of F ( a), which is also an extension of F . This leads us to the following definition. 1.6.4. Definition. If F is a field and K is an extension of F , say that K is a simple quadratic extension of F if there is a positive number a in F such that a / F and K = F ( a). Say that a field F is a quadratic extension of Q if there are fields F0,...,Fn such that: (a) F0 = Q and Fn = F (b) for 1 k n, Fk is a simple quadratic extension of Fk−1 Say that F is a quadratic extension of Q of degree n if there are n + 1 fields F0,F1,...,Fn as just described and no such chain of simple quadratic extensions can be found with fewer than n + 1 fields. It is not difficult to show that the intersection of any collection of fields contained in R is also a field (Exercise 5). So if we are given two fields F and K, we can talk about the field generated by F and K as the smallest field containing F K, that is, the field {L : L is a field and L F K}. If F and K are quadratic extensions of Q, then it is true but not so clear that the field they generate is also a quadratic extension of Q. 1.6.5. Proposition. If F and K are quadratic extensions of Q of degrees n and m, respectively, then the field generated by F and K is a quadratic extension of Q of degree at most n + m. More generally, if F1,...,Fn are quadratic extensions of Q, then the field generated by F1,...,Fn is a qua- dratic extension of Q.
Previous Page Next Page