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18 1. Trisecting Angles 1.5.9. Trisection Problem. For each constructible angle is there a con- structible angle with one-third the radians? Exercises 1. Show that if (a, b) is a constructible point, then so are (−a, b) and (a, −b). Use this to complete the proof of Lemma 1.5.4. 2. Prove Claim 5 in Theorem 1.5.6. (Hint: you might first prove that if b ∈ K and b = 0, then 1/b ∈ K. After this you can use Claim 4.) 3. Prove that (2(2 √ 2)1/2)1/2 ∈ K. 4. Is 2 1 8 constructible? 5. What is the smallest integer n so that √ 2 ∈ Pn? 6. How many points are in P3? How many straight lines are in L3? 7. If C is a constructible circle containing the constructible point P , show that the line through P that is tangent to C is a constructible line. 1.6. Quadratic field extensions Here we need the concept of a field, though we’ll avoid the abstract defini- tion. (See A.3.1 below.) For our purposes we only examine fields that are contained in the real numbers, R. Thus our working definition of a field will be the following. A field is a subset F of R satisfying the following properties: (a) 0 and 1 belong to F (b) if a, b ∈ F , then a + b, a − b, and ab ∈ F (c) if a, b ∈ F and b = 0, then a/b ∈ F . Of course Q is a field as is R. Also Theorem 1.5.6 says that K is a field. The next result is a straightforward consequence of the axioms. 1.6.1. Proposition. Every field contains the rational numbers. 1.6.2. Definition. If F is a field, then an extension of F is a field K that contains F . So R and K are both extensions of Q, and R is an extension of K. 1.6.3. Proposition. If F is a field and a is a positive number in F , then F ( √ a) ≡ {x + y √ a : x, y ∈ F } is an extension of F .