20 1. Trisecting Angles Proof. By definition there are fields F0,...,Fn with F0 = Q, Fn = F , and for 1 ≤ k ≤ n there is a positive element ak in Fk−1 such that Fk = Fk−1 √ ak . Similarly there are fields K0,...,Km such that K0 = Q, Km = K, and for 1 ≤ j ≤ m there is a positive element bj−1 in Kj−1 such that Kj = Kj−1 ( bj . Form the fields L0 = Q,L1 = F1,...,Ln = Fn = F, Ln+1 = Ln (√ b1 ) , . . . , Ln+m = Ln+m−1 (√ bm ) . It is left to the reader to show that Ln+m is the field generated by F and K, though it is possible that not all the fields in this sequence are proper extensions of their immediate predecessor. The proof of the more general statement follows by induction. We might observe that K has no proper quadratic extensions since the square root of any constructible number is constructible (1.5.8). We’ll use this same fact to prove half of the following theorem. 1.6.6. Theorem. If F is a quadratic extension of Q, then F ⊆ K. Con- versely, if a ∈ K, then there is a field F which is a quadratic extension of Q such that a ∈ F . In fact, if K is any field contained in K and a ∈ K with a 0, then √ a ∈ K, so K ( a) ⊆ K. The proof that every quadratic extension of Q is contained in K can now be completed by induction. The proof of the converse requires some lemmas and notation. Since this half of the theorem is not needed to show that there are angles that cannot be trisected, the impatient reader can temporarily skip the remainder of this section and return at his/her leisure. 1.6.7. Definition. If F is a field, a point in the plane is called an F -point if its coordinates are numbers in F . A line is called an F -line if it passes through two F -points. A circle is called an F -circle if its center is an F -point and its radius is the distance between two F -points. 1.6.8. Lemma. Let F be a field. (a) y = mx + b is the equation of an F -line if and only if m, b ∈ F . (b) (x − h)2 + (y − k)2 = r2 is the equation of an F -circle if and only if h, k, r2 ∈ F . Proof. (a) (0,b) and (1,m + b) lie of the straight line so if m, b ∈ F , then these are F -points. Conversely, assume that y = mx + b is an F -line. So there are F -points (x1,y1) and (x2,y2) that lie on this line. Standard algebra tells us that m = y2 − y1 x2 − x1 ∈ F. Also b = y1 − mx1 ∈ F .

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