Abstract Algebra Theory and Applications - Computer Science ...

338 CHAPTER 19 FIELDSlength αβ. A similar construction can be made if β < 1. We will leave it asan exercise to show that the same triangle can be used to construct α/β forβ ≠ 0.□DβBA1αxCEFigure 19.1. Construction of productsLemma 19.22 If α is a constructible number, then √ α is a constructiblenumber.Proof. In Figure 19.2 the triangles △ABD, △BCD, and △ABC aresimilar; hence, 1/x = x/α, or x 2 = α.□BxA1 αDCFigure 19.2. Construction of rootsBy Theorem 19.21, we can locate in the plane any point P = (p, q) thathas rational coordinates p and q. We need to know what other points canbe constructed with a compass and straightedge from points with rationalcoordinates.Lemma 19.23 Let F be a subfield of R.