provide full solution with the given theorems
a) Formalizing proof of Theorem 12.3.12. First use the language of 'definition by recursion' to formalize the Definition 12.2.1 for what it means by a number to be Constructible. Then, use the framework of structural induction to furnish a formal proof for Theorem 12.3.12 (of course the details of the proof and the supporting theorems are all coming out of the textbook proof.) b) Carefully read the statement and the proof of Theorem 12.3.21. Would the theorem still hold if we assume the coefficients of the polynomial, a and b are not necessarily rational, but they are from the subfield F? Explain. ) Prove or disprove: if a cubic polynomial with rational coefficients has a constructible root then all its real roots must be constructible. d) Show that the cubic polynomial ' + r + v3 cannot have rational roots. Does it have a constructible root? (Hint: consider Ex 16, and read its solution in the solutions manual.) e) Carefully read Theorem 12.3.22 and its proof. Give a counterexample for the following claim, and point out what is wrong with the argument: (not a) Theorem: Suppose a cubic polynomial with coefficients in Q(v3) has a constructible root. Then it must have a rational root. Proof: choose a shortest tower of fields where a root of the polynomial is found. If this tower is not Q then it is F(vs) for some s, that is the root is r = a + bys. Now By the argument in theorem 12.3.21, we must have a root in F, which is shorter than F(vs). This contradiction proves the shortest tower in which there is a root must be Q. Theorem 12.3.12. The field of constructible numbers is the same as the field of surds Proof. We already showed that every surd is constructible (Theorem 12.3.3). On the other hand, Theorems 12.3.7, 12.3.8, 12.3.9, 12.3.10, and 12.3.11 show that starting with surd points in the plane and doing geometric constructions produces only surd points. Thus, every constructible point in the plane has surd coordinates. Since every constructible number is a coordinate of a constructible point in the plane (Theorem 12.3.5), it follows that every constructible number is a surd. This characterization of the constructible numbers is the key to the proof that certain angles cannot be trisected. One of the relationships between constructible angles and constructible numbers can be obtained using the trigonometric function cosine, as follows. We restrict the discussion to acute angles (i.e., angles less than a right angle) simply to avoid having to describe several cases. Definition 12.2.1. A real number is constructible if the point corresponding to it on the number line can be obtained from the marked points 0 and 1 by performing a finite sequence of geometric constructions in the plane using only a straightedge and compass. Theorem 12.3.21. If a + br is in F (vr) and is a root of a polynomial with rational coefficients, then a - bar is also a root of the polynomial. 152 12 Constructibility Proof. Suppose that an (atb /r)" tan-1(atbr)" + ... tai(atbr) tao=0. Then, an ( a + br) " + an-1 (atbvr)" + ...+ a(atbvr) +ao=0=0 Since each of the coefficients ak is rational, ak = ak, for every k. Using this fact and the facts that the conjugate of a sum is the sum of the conjugates and the conjugate of a product is the product of the conjugates (Theorem 12.3.20), it follows that an (a + br) "tan-1 (a + but)" -+.. .tai(a+ but)tao= 0. Thus, a-bVT = a + bvr is also a root of the polynomial. Theorem 12.3.22. If a cubic equation with rational coefficients has a constructible root, then the equation has a rational root. Proof. Dividing through by the leading coefficient, we can assume that the coeffi- cient of x' is 1. Then, by Theorem 12.3.18, the sum of the three roots of the cubic equation is the negative of the coefficient of x2, and is therefore rational. We first show that if the equation has a root in any F (vr), then it has a root in F. To see this, suppose the equation has a root in F (vr) of the form a + bur with b # 0. Then, by Theorem 12.3.21, the conjugate a - bvr is also a root. If r3 is the third root and s is the sum of all three roots, then s = 13 + (a +bur) +(a-b /r) = 3 + 2a. Thus, r3 = s - 2a. Since F contains all rational numbers and s is rational, s is in F . Since a is also in F , it follows that the root r3 is in F itself The preliminary result obtained in the previous paragraph allows us to prove the theorem, as follows. If the polynomial has a constructible root, then, since every constructible number is a surd (Theorem 12.3.12), the root is in a field that occurs at the end of a tower. Consider the field at the end of the shortest tower that contains any root of the given cubic equation. We claim that field is Q. To see this, simply note that if that field was F (vr), the previous paragraph would imply that there is a root in F, which would be at the end of a shorter tower than F (vr) is in. Hence, that field is Q. Thus, the equation has a rational root. We can now prove that an angle of 20 cannot be constructed
