MAT246H1S Lec0101 Burbulla

Chapter 12: Constructibility

Chapter 12 Lecture Notes Constructibility

Winter 2019

Chapter 12 Lecture Notes Constructibility MAT246H1S Lec0101 Burbulla

Chapter 12: Constructibility

Chapter 12: Constructibility 12.1: Constructions With Straight Edge and Compass 12.2: Constructible Numbers 12.3: Surds

This section illustrates how to perform six basic constructions with straight edge and compass, namely how to construct 1. the perpendicular bisector of a given line segment, 2. the bisector of a given angle, 3. a copy of a given line segment, 4. a copy of a given angle, 5. the sum of two given angles, and any multiple of an angle, 6. the division of a given line segment into n equal parts. The constructions will be illustrated on the board, not given in the notes.

Chapter 12 Lecture Notes Constructibility MAT246H1S Lec0101 Burbulla

12.1: Constructions With Straight Edge and Compass Chapter 12: Constructibility 12.2: Constructible Numbers 12.3: Surds Bisectors

Deﬁnition: A perpendicular bisector of a line segment is a line that is perpendicular to the given line segment and goes through the middle of the line segment.

Theorem 12.1.2: given any line segment, its perpendicular bisector can be constructed.

Deﬁnition: An angle bisector of a given angle is a line from the vertex of the given angle that divides it into two equal subangles.

Theorem 12.1.4: given any angle, its bisector can be constructed.

Theorem 12.1.5: any given line segment can be copies using only a straight edge and compass.

Theorem 12.1.6: any given angle can be copied using only straight edge and compass.

Corollary 12.1.7: if angles α and β can be constructed, then (i) the angle α + β can be constructed. (ii) then angle n α can be constructed, for any natural number n.

Chapter 12 Lecture Notes Constructibility MAT246H1S Lec0101 Burbulla

12.1: Constructions With Straight Edge and Compass Chapter 12: Constructibility 12.2: Constructible Numbers 12.3: Surds Dividing a Line Segment Into n Equal Parts

Theorem 12.1.8: given any line segment and any natural number n, the line segment can be divided into n equal parts using only straight edge and compass.

Start with an arbitrary line segment and label one end of it 0 and the other end of it 1, and consider the number line passing through the given line segment. 0 1 X

r r r

Deﬁnition: a real number x is constructible if the point corresponding to it on the number line, X , can be obtained from the marked points 0 and 1 by performing a ﬁnite sequence of constructions using only straight edge and compass.

The goal of Sections 2.2 and 12.3 is to ﬁnd and describe algebraically all numbers that are constructible.

Chapter 12 Lecture Notes Constructibility MAT246H1S Lec0101 Burbulla

12.1: Constructions With Straight Edge and Compass Chapter 12: Constructibility 12.2: Constructible Numbers 12.3: Surds

Z and Q Are Constructible

Theorem 12.2.2: every integer is constructible.

Theorem 12.2.3: every rational number is constructible.

Theorem 12.2.4: if a is constructible then −a is constructible. Theorem 12.2.5: the sum of two constructible numbers is constructible. Theorem 12.2.6: if a and b are positive constructible numbers, a then is constructible. b Corollary 12.2.7: if a and b are constructible numbers, then ab is a constructible, and if b 6= 0, then is constructible. b Proof: if a and b are both positive then c = 1/b is constructible, by Th. 12.2.6 using a = 1, and then a/c = ab is constructible, by Th. 12.2.6. If a and b are both negative, apply above to |ab| = ab and to |a/b| = a/b, if b 6= 0. If only one of a, b is negative, then ab = −|ab|, and if b 6= 0, then a/b = −|a/b|, which are both constructible. Chapter 12 Lecture Notes Constructibility MAT246H1S Lec0101 Burbulla

12.1: Constructions With Straight Edge and Compass Chapter 12: Constructibility 12.2: Constructible Numbers 12.3: Surds

What Is A Subfield of R? The following is a key definition for the rest of this chapter. Definition: a subfield of R, of simply a field, is a set F of real numbers such that 1. 0 and 1 are in F. 2. if x, y ∈ F then x + y ∈ F and xy ∈ F. 3. if x ∈ F then −x ∈ F. 1 4. if x ∈ F and x 6= 0, then ∈ F. x Examples: both Q and R are subfields of R. Theorem 12.2.9: if F is any subfield of R, then Q ⊂ F ⊂ R. That is, Q is the “smallest” subfield of R, and R is the “largest” subfield of R. Proof: follows from properties 1, 2, 3 and 4 applied to x, y ∈ N. Chapter 12 Lecture Notes Constructibility MAT246H1S Lec0101 Burbulla 12.1: Constructions With Straight Edge and Compass Chapter 12: Constructibility 12.2: Constructible Numbers 12.3: Surds

The Constructible Numbers Form a Subﬁeld of R.

Theorem 12.2.10: the set of all constructible numbers is a subﬁeld of R. Proof: follows from Theorems 12.2.2 to Corollary 12.2.7.

Question: what does the subﬁeld of constructible numbers look like?

Chapter 12 Lecture Notes Constructibility MAT246H1S Lec0101 Burbulla

12.1: Constructions With Straight Edge and Compass Chapter 12: Constructibility 12.2: Constructible Numbers 12.3: Surds Example 1 √ √ Q( 2) = {a + b 2 | a, b ∈ Q} is a subﬁeld of R. Proof: a√= 0, b = 0 gives√ you 0; a = 1, b = 0 gives you 1. −(a + b 2) = −a − b 2, and −a, −b are both in Q. Closed under addition and multiplication: √ √ √ (a1 + b1 2) + (a2 + b2 2) = (a1 + a2) + (b1 + b2) 2;

√ √ √ (a1 + b1 2)(a2 + b2 2) = (a1a2 + 2b1b2) + (a1b2 + a2b1) 2. √ Finally, if a + b 2 6= 0, then √ 1 a − b 2 √ = , a + b 2 a2 − 2b2 √ and a2 − 2b2 6= 0, since 2 is not rational.

Theorem 12.2.12: let F be any subﬁeld of and suppose that r √ R is a positive number in F. If r is not in F and √ √ F( r) = {a + b r | a, b ∈ F}, √ then F( r) is a subﬁeld of R. Proof: mimic Example 1.

Definition: if F is a subfield of and r is a positive number in F √ R such that r is not in F, then the field √ F = {a + b r | a, b ∈ F} √ is the field obtained by adjoining r to F, and is called the √ extension of F by r.

Chapter 12 Lecture Notes Constructibility MAT246H1S Lec0101 Burbulla

12.1: Constructions With Straight Edge and Compass Chapter 12: Constructibility 12.2: Constructible Numbers 12.3: Surds Example 2

√ √ √ √ Since 5 is not in Q( 2), the extension of Q( 2) by 5 is √ √ √ √ √ {a+b 5 | a, b ∈ Q( 2)} = {c+d 2+(e+f 2) 5 | c, d, e, f ∈ Q}. √ √ It can also be written as [Q( 2)]( 5).

Theorem 12.2.15: if r is a positive constructible number, then √ r is constructible.

Thus √ I Every number in Q( 2) is constructible. √ If r is rational but r is irrational, then every number in I √ Q( r) is constructible. If F is a subﬁeld of and every number in F is constructible, I R √ then every number in F( r) is constructible, if r is a positive √ number in F but r is not in F.

Chapter 12 Lecture Notes Constructibility MAT246H1S Lec0101 Burbulla

12.1: Constructions With Straight Edge and Compass Chapter 12: Constructibility 12.2: Constructible Numbers 12.3: Surds A Tower of Fields

The following definition is somewhat complicated but crucial for describing algebraically what constructible numbers ‘look’ like. Definition: a finite sequence F0, F1,..., Fn of subfields of R such that F = , and for each i, from 1 to n, there is a positive 0 Q √ √ number ri ∈ Fi−1 such that ri ∈/ Fi−1 but Fi = Fi−1( ri ), is called a tower of fields. We have

F0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Fn−1 ⊂ Fn. From Example 2, a short tower of ﬁelds is √ √ √ √ √ F0 = Q, F1 = F0( 2) = Q( 2), F2 = F1( 5) = [Q( 2)]( 5), or equivalently, √ √ √ Q ⊂ Q( 2) ⊂ [Q( 2)]( 5).

Definition: a surd is a number that is in some field that is in a tower of fields. That is, x ∈ R is a surd if there exists a tower of fields F0 ⊂ F1 ⊂ · · · ⊂ Fn−1 ⊂ Fn

such that x ∈ Fn. √ √ √ √ √ √ Example: x = (2 + 2) + (5 − 2) 5 = 2 + 2 + 5 5 − 10 is a surd, since √ √ √ √ √ (2 + 2) + (5 − 2) 5 ∈ [Q( 2)]( 5),

and √ √ √ Q ⊂ Q( 2) ⊂ [Q( 2)]( 5).

Chapter 12 Lecture Notes Constructibility MAT246H1S Lec0101 Burbulla

12.1: Constructions With Straight Edge and Compass Chapter 12: Constructibility 12.2: Constructible Numbers 12.3: Surds The Set of All Surds Is a Field

Theorem 12.3.2: the set of all surds is a subﬁeld of . Moreover, √ R if r is a positive surd, then r is also a surd. √ √ Proof: suppose x ∈ F( rm), y ∈ F( sn) with √ √ √ Q ⊂ F( r1) ⊂ F( r2) · · · ⊂ F( rm),

and √ √ √ Q ⊂ F( s1) ⊂ F( s2) · · · ⊂ F( sn). Then x and y are both in the tower of fields produced by adjoining r , r ,..., r , s , s ,..., s to F = . Finally, if r is in some field 1 2 m 1 2 n √ 0 Q √ √ F then it is in a tower. If r ∈ F, then r is a surd; if r ∈/ F, √ √ then r ∈ F( r) which is in a tower of fields that has one more field than the tower leading to F.

Theorem 12.3.3: Every surd is constructible.

Proof: every tower of fields starts at Q, and every rational number is constructible, by Theorem 12.2.3. If every number in F is √ constructible then every number in F( r) is also constructible, by Theorem 12.2.15. Thus every surd is constructible. In fact, the converse is also true: every constructible number is a surd. To prove this, and we will only outline the proof, we will have to work in the plane, since all straight edge and compass constructions occur in the plane. To this end: Definition: the point (x, y) is constructible if the point can be obtained from the points (0, 0) and (1, 0) by performing a finite sequence of constructions with straight edge and compass.

Chapter 12 Lecture Notes Constructibility MAT246H1S Lec0101 Burbulla

12.1: Constructions With Straight Edge and Compass Chapter 12: Constructibility 12.2: Constructible Numbers 12.3: Surds The Surd Plane

Theorem 12.3.5: the point (x, y) is constructible if both of the coordinates x and y are constructible numbers.

Deﬁnition: The surd plane is the set of all points (x, y) in the xy-plane such that the coordinates x and y are both surds. Comments: 1. Since every surd is constructible we know every point in the surd plane is constructible. 2. It remains to show that every constructible point is in the surd plane.

To this end we should prove Theorems 12.3.7, 12.3.8, 12.3.9, 12.3.10 and 12.3.11. Instead, we shall just list them. Check the details in the book, if you like.

Theorem 12.3.7: if a line goes through two points in the surd plane, then there is an equation for that line with surd coefficients. Theorem 12.3.8: a circle whose centre is in the surd plane and whose radius is a surd has an equation in which the coefficients are all surds. Theorem 12.3.9: the point of intersection of two distinct lines that each go through two points in the surd plane is itself in the surd plane. Theorem 12.3.10: the points of intersection of a line that has an equation with surd coefficients and a circle that has an equation with surd coefficients lie in the surd plane. Theorem 12.3.11: the points of intersection of two distinct circles that have equations with surd coefficients lie in the surd plane.

Chapter 12 Lecture Notes Constructibility MAT246H1S Lec0101 Burbulla

12.1: Constructions With Straight Edge and Compass Chapter 12: Constructibility 12.2: Constructible Numbers 12.3: Surds Every Constructible Number Is A Surd

Theorem 12.3.12: the ﬁeld of constructible numbers is the same as the ﬁeld of surds. Proof: by Theorem 12.3.3, we know every surd is constructible. Theorems 12.3.7, 12.3.8, 12.3.9, 12.3.10 and 12.3.11 show that the only constructible points in the plane, with straight edge (lines) and compass (circles), are points with surd coordinates. Thus every constructible number is a surd. Comments: now we can answer some questions about constructibility algebraically. For example, we shall be able to answer the three classic unsolved questions of ancient Greek geometry: 1. Can an arbitrary angle be trisected? 2. Can the cube be doubled? 3. Can the circle be squared?

Theorem 12.3.13: the acute angle θ is constructible with a straight edge and compass if and only if cos θ is a constructible number. Theorem 12.3.14: an angle of 60◦ is constructible . 1 Proof: cos 60◦ = , which is constructible. 2 Corollary 12.3.15: angles of 30◦, 15◦, 45◦, 75◦ are constructible. Proof: bisect 60◦ to get 30◦; bisect 30◦ to get 15◦; add 30◦ and 15◦ to get 45◦; add 30◦ and 45◦ to get 75◦. Question: is an angle of 20◦ constructible? The answer is no, but we will need some trigonometry to prove it.

Chapter 12 Lecture Notes Constructibility MAT246H1S Lec0101 Burbulla

12.1: Constructions With Straight Edge and Compass Chapter 12: Constructibility 12.2: Constructible Numbers 12.3: Surds Two Useful Trigonometric Identities

Theorem 12.3.16: for any angle θ,

cos(3θ) = 4 cos3 θ − 3 cos θ, and sin(3θ) = 3 sin θ − 4 sin3 θ.

Proof: we shall use De Moivre’s Theorem.

cos(3θ) + i sin(3θ) = (cos θ + i sin θ)3 = cos3 θ + 3 cos2 θ i sin θ − 3 cos θ sin2 θ − i sin3 θ

So cos(3θ) = cos3 θ − 3 cos θ sin2 θ = cos3 θ − 3 cos θ (1 − cos2 θ), sin(3θ) = 3 cos2 θ sin θ − sin3 θ = 3(1 − sin2 θ) sin θ − sin3 θ, and the results follow.

If the angle 20◦ were constructible, then x = cos 20◦ would be constructible. But by the identity on the previous slide, 1 = cos 60◦ = 4 cos3 20◦−3 cos 20◦ ⇔ 1 = 8x3−6x ⇔ 8x3−6x−1 = 0. 2 So 20◦ is constructible if and only if the cubic polynomial 8x3 − 6x − 1 has a constructible root. We shall show this is impossible, so 20◦ is not constructible, and so trisecting a general angle with straight edge and compass is impossible. But to do this we shall need some more theory about cubic polynomials. In particular, we shall prove that if a cubic polynomial with rational coeﬃcients has a constructible root, then it has a rational root. But the above polynomial has no rational root (why?) so x can’t be constructible.

Chapter 12 Lecture Notes Constructibility MAT246H1S Lec0101 Burbulla

12.1: Constructions With Straight Edge and Compass Chapter 12: Constructibility 12.2: Constructible Numbers 12.3: Surds Roots of Cubics

Theorem 12.3.18: if the roots of the cubic equation

x3 + bx2 + cx + d = 0

are r1, r2 and r3, then b = −(r1 + r2 + r3). Proof: expand the product of the three factors of the polynomial:

3 2 (x−r1)(x−r2)(x−r3) = x −(r1+r2+r3)x +(r1r2+r2r3+r3r1)x−r1r2r3;

3 2 thus (x − r1)(x − r2)(x − r3) = x + bx + cx + d if and only if

b = −(r1 + r2 + r3).

Chapter 12 Lecture Notes Constructibility MAT246H1S Lec0101 Burbulla 12.1: Constructions With Straight Edge and Compass Chapter 12: Constructibility 12.2: Constructible Numbers 12.3: Surds √ Conjugates in F( r) √ √ √ Deﬁniton: if a + b r ∈ F( r), the conjugate of a + b r is √ √ a + b r = a − b r. Theorem 12.3.20: the conjugate of a sum is the sum of the conjugates and the conjugate of a product is the product of the conjugates. Proof: just calculate: √ √ √ √ a + b r + c + d r = (a + c) + (b + d) r = a + c − (b + d) r, √ √ √ √ which is equal to a + b r + c + d r = a − b r + c − d r. And √ √ √ (a + b r)(c + d r) = ac + bdr + (bc + ad) r √ = ac + bdr − (bc + ad) r, √ √ √ √ and a + b r c + d r = (a − b r)(c − d r) √ = ac + bdr − (bc + ad) r.

Chapter 12 Lecture Notes Constructibility MAT246H1S Lec0101 Burbulla

12.1: Constructions With Straight Edge and Compass Chapter 12: Constructibility 12.2: Constructible Numbers 12.3: Surds Conjugate Roots of a Polynomial with Rational Coeﬃcients √ √ Theorem 12.3.21: if a + b r is in F( r) and is a root of a √ √ polynomial with rational coeﬃcients, then a + b r = a − b r is also a root of the polynomial.

Proof: use Theorem 12.3.20 and the fact that if q ∈ Q then n n−1 q = q. Let f (x) = anx + an−1x + ··· + a1x1 + a0, with ai ∈ Q. Then

√ √ n √ f (a + b r) = 0 ⇒ an(a + b r) + ··· + a1(a + b r) + a0 = 0 √ √ n ⇒ an(a + b r) + ··· + a1(a + b r) + a0 = 0 √ n √ ⇒ an(a + b r) + ··· + a1(a + b r) + a0 = 0 √ ⇒ f (a + b r) = 0 √ ⇒ f (a − b r) = 0

Theorem 12.3.22: if a cubic equation with rational coefficients has a constructible root, then the equation has a rational root. Proof: if necessary, divide the cubic by the coefficient of x3 to get a cubic polynomial of the form f (x) = x3 + Bx2 + Cx + D, with B, C, D ∈ . Let the roots of f be r , r , r . By Theorem Q 1 2 3 √ 12.3.18, r + r + r = −B. Assume that r = a + b r, with 1 2 3 √ 1 b 6= 0, is constructible: then r ∈ F( r) for some field extension √ √1 F( r), with a, b, r ∈ F but r ∈/ F. By Theorem 12.3.21 √ √ r1 = a − b r is also a root of f (x); suppose r2 = a − b r. Then √ √ −B = r1 + r2 + r3 = a + b r + a − b r + r3 = 2a + r3,

which implies r3 ∈ F. (Because B ∈ Q ⊂ F, and 2a ∈ F.) Chapter 12 Lecture Notes Constructibility MAT246H1S Lec0101 Burbulla

12.1: Constructions With Straight Edge and Compass Chapter 12: Constructibility 12.2: Constructible Numbers 12.3: Surds

Thus f (x) has a root in F. That is, (*): if f (x) has a root in the √ √ ﬁeld extension F( r), then it has a root in F. Suppose F( r) is the last ﬁeld in the tower √ Q = F0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Fn−1 ⊂ Fn = F( r).

What is the smallest ﬁeld in this tower that contains a root of f (x)? If it is Fi then by (*), there would be a root in Fi−1, which means that the smallest ﬁeld in the tower containing a root of f (x) must be F0 = Q itself. That is, f (x) has a rational root.

Theorem 12.3.23: an angle of 20◦ cannot be constructed with straight edge and compass. Proof: from previous considerations we know that if cos 20◦ were constructible the cubic polynomial

f (x) = 8x3 − 6x − 1

would have a constructible root, and therefore by Theorem 12.3.22, it would have a rational root. By the Rational Roots Theorem, the only rational roots could be 1 1 1 x = ±1, ± , ± , ± , 2 4 8 none of which satisfy f (x) = 0. The angle 20◦ is not constructible.

Chapter 12 Lecture Notes Constructibility MAT246H1S Lec0101 Burbulla