Abstract Algebra Theory and Applications - Computer Science ...

340 CHAPTER 19 FIELDSbe the equations of two circles, where d i , e i , and f i are in F for i = 1, 2.These circles have the same intersection as the circleand the linex 2 + y 2 + d 1 x + e 1 x + f 1 = 0(d 1 − d 2 )x + b(e 2 − e 1 )y + (f 2 − f 1 ) = 0.The last equation is that of the chord passing through the intersection pointsof the two circles. Hence, the intersection of two circles can be reduced tothe case of an intersection of a line with a circle.Considering the case of the intersection of a line and a circle, we mustdetermine the nature of the solutions of the equationsax + by + c = 0x 2 + y 2 + dx + ey + f = 0.If we eliminate y from these equations, we obtain an equation of the formAx 2 + Bx + C = 0, where A, B, and C are in F . The x coordinate of theintersection points is given byx = −B ± √ B 2 − 4AC2Aand is in F ( √ α ), where α = B 2 − 4AC > 0. We have proven the followinglemma.Lemma 19.24 Let F be a field of constructible numbers. Then the pointsdetermined by the intersections of lines and circles in F lie in the fieldF ( √ α ) for some α in F .Theorem 19.25 A real number α is a constructible number if and only ifthere exists a sequence of fieldsQ = F 0 ⊂ F 1 ⊂ · · · ⊂ F ksuch that F i = F i−1 ( √ α i ) with α ∈ F k . In particular, there exists an integerk > 0 such that [Q(α) : Q] = 2 k .Proof. The existence of the F i ’s and the α i ’s is a direct consequence ofLemma 19.24 and of the fact that[F k : Q] = [F k : F k−1 ][F k−1 : F k−2 ] · · · [F 1 : Q] = 2 k .□