Delian Problem and Other Construction Problems

Home , Pierre Wantzel

Delian problem and other construction problems

LIU Rui Ling

(19220642)

A thesis submitted in partial fulfillment of the requirements for the degree of

Bachelor of Science (Honours)

in Mathematics and Statistics

Hong Kong Baptist University

14 January 2021

Acknowledgements Thanks to my supervisor, Dr. Sun Pak Kiu provided some useful reading materials for me so that I have a clear idea of how completing this thesis. Also, I am much obliged to Dr. Sun for his friendly reminder and support.

______Signature of Student

LIU Rui Ling Student Name

Department of Mathematics Hong Kong Baptist University

Date: 14 January 2021

Delian problem and other construction problems

LIU Rui Ling (19220642)

Department of Mathematics

Abstract

In this thesis, I will discuss the problem of duplication of a cube, which is one of the three geometrical construction problems in ancient Greek mathematics. The problem of doubling the cube refers to whether we can use a straightedge with no scale and compass to construct a cube so that its volume is equal to twice the original one. In chapter one, I will introduce the history of the problem of doubling the cube. In chapter two, I will discuss the straightedge and compass construction. Therefore, I will explain that how to construct numbers by using straightedge and compass construction. In chapter three, I will discuss the fields. In chapter four, I will discuss the field extensions, including the vector space, the field extensions and the algebraic extension. In chapter five, I will prove that why we cannot solve the problem of doubling the cube by using the straightedge and compass construction.

Table of Contents Acknowledgements ...... 1 Abstract ...... 2 Chapter 1The history of the problem of doubling the cube ...... 4 Chapter 2 Constructible Numbers ...... 5 2.1 The straightedge and compass construction ...... 5 Chapter 3 Fields ...... 10 Chapter 4 The field extensions ...... 11 4.1 The vector space ...... 11 4.2 The field extensions ...... 13 4.3 The algebraic extension ...... 15 Chapter 5 Doubling the cube ...... 17 Chapter 6 Conclusion ...... 19 References ...... 20

Chapter 1 The history of the problem of doubling the cube Delian problem, also called the problem of doubling the cube, is one of the three geometric construction problems raised by the ancient Greeks 2,400 years ago. The problem of doubling the cube is to construct a cube which its volume is equal to twice the original one. The reason why this question is difficult to solve is that there are restrictions on the drawing tools. The ancient Greeks emphasized that only straightedge with no scales and compasses could be used for geometric drawing. There is a myth about the origin of the problem of doubling the cube. [1] The myth is that a plague broke out in the island of Delos. A quarter of the population of Delos died of the plague. The islanders went to the temple to ask for Apollo's will. The oracle replied that if they want to stop the plague, they should double the volume of altar in the Apollo’s temple. The islanders doubled the length of each side. The new altar was finished, and the plague continued to spread. They felt confused. They asked the famous philosopher and mathematician Plato about the reason why the oracle was not working. Plato pointed out the problem. They doubled each side of the altar, so that the volume would become eight times the original size instead of doubling. At this time, everyone suddenly realized. So, if the volume is to be doubled, how many should the length of each side increase? This is the problem of doubling the cube among the three famous construction problems in ancient Greece. Since then the problem of doubling the cube also called the Delian problem. Since the appearance of the problem of the doubling the cube, it has attracted many mathematicians to study. In 1837, a French mathematician, Pierre Wantzel proved that we could not use the straightedge and compass construction to solve the Delian problem.

Chapter 2

Constructible Numbers

2.1 The straightedge and compass construction

Straightedge and compass construction is a mathematical problem that originated in ancient Greece. Only compasses and straightedges can be used to solve a given problem in a limited number of steps. In the straightedge and compass construction, the definition of straightedge and compass as below: A straightedge is unmarked, it can only be used for drawing a line segment through any two constructed points. A compass is used for drawing a circle with a constructed point O as the center and a constructed line segment r (r>0) as the radius. For two arbitrary points, point A and point B, we define |AB | as their distance. First, finding two points A and B. The line determined by these two points is the x- axis. Second, drawing a circle with A as the center and |AB | as the radius by using a compass. Final, drawing a circle with B as the center and |AB | as the radius by using a compass. Two circles which we had drawn intersect at point C and point D, and the straight line determined by these two points is the y-axis. The two line segments, that are AB and CD intersect at a point O, which is the origin of the coordinate system. We define the length of |OA| as 1 unit, and the coordinate of O is (0,0). So, we can draw a line segment is 1 unit of length at the beginning. See Figure 2.1.

Definition 2.1.1 The constructible number: The constructible number refers the real number that can be drawn in a limit number of steps by using a straightedge. In the case of a given |α| (α∈ ℝ) unit length, if the line segment could be drawn by using a straightedge, then the number α is a constructible. [2]

All the constructing steps can be reduced to the following five basic steps: Definition 2.1.2 The constructible point: The constructible point refers to as below: (a) An intersection point of 2 non-parallel constructible segments, or (b) An intersection point of 1 constructible segment and a constructible circle, or (c) An intersection point of 2 constructible circles. Definition 2.1.3 The constructible segment: The constructible segment refers to a line segment which two endpoints are constructible points. Definition 2.1.4 The constructible circle: The constructible circle refers to a circle with a constructible point as the center and a constructible segment as the radius. The length of each line segment is equivalent to a real number. Here we assume that "given a positive real number" means "given a line segment, and the length is equivalent to a given positive real number. Therefore, given two positive real numbers, we can construct the sum of the two numbers and difference. Given two positive real numbers and a line segment of length 1unit, then the product, quotient and square root of the two numbers could be constructed. So, for a given constructed line segment, we can obtain a constructible number by the five mathematical operation: addition, subtraction, multiplication, division and taking a square root. Let α, β>0, the five operation as show below: 1. α + β For two given line segments, line segment AB and line segment CD which the length of α and β respectively. Constructing a circle with point B as the center and line segment CD as the radius by a compass. Constructing a line segment AB, and the two intersection points of line segment AB and a constructed circle are point M and N respectively. See Figure 2.1.2.

2. α - β Two numbers that are α-β and α+β have a similar construction, but the line segment of CD is constructed in the opposite side. See Figure 2.1.3.

3. α x β For three given line segments, OP and OA which the length of α and βrespectively, and a line segment of length 1unit. Constructing a line segment OA with a length of β by using a compass. Let |OR|=1 unit, and point R lies on the line segment OA. Constructing a line l through point O and do not contain OA. Let |OP|=α, and a point P in the line l. Using the straightedge to construct a line segment RP and construct a line segment AQ from point A, such that a line segment PR and a line segment AQ is a pair of parallel lines, and the line segment AQ intersects the line l at point Q. Now, we have a pair of similar triangles, the triangle OPR is similar to the triangle OQA. According to the properties of similar triangle, we have 1 β = . 훼 |푂푄| So |OQ|=αβ. See Figure 2.1.4.

4. α / β Two numbers that are α/β and αβ have a similar construction. For three given line segments, OQ and OA which the length of α and β (whereβ≠0) respectively, and a line segment of length 1unit. Constructing a line segment OA with a length of β by using a compass. Let |OR|=1 unit, and point R lies on the line segment OA. Constructing a line l through point O and do not contain OA. Let |OQ|=α, and point Q lies on the line l. Using the straightedge to construct a line segment AQ and a line segment RP from a point R, such that the line segment AQ is parallel to the line segment RP, and the line segment RP intersects the line l at point P. Now, we have a pair of similar triangles, the triangle OPR is similar to the triangle OQA. According to the properties of similar triangle, we have |푂푃| 훼 = . 1 β 훼 So |OP|= . See Figure 2.1.5. β

√α For a given line segment EA with the length of α, and a line segment of length 1unit. Extending the line segment AE to a point O such that the line segment OE with the length of 1 unit. Using a compass to find the midpoint of the line segment OA, and construct a circle with the midpoint as the center and line segment OA as the diameter. Using a compass to a perpendicular line ER to the line segment OA from point E. Now, we have a pair of similar triangles, the triangle EOR is similar to the triangle ERA. According to the properties of similar triangle, we have

|퐸푅| 1 = . 훼 |퐸푅| So, |ER|=√α. See Figure 2.1.6.

Chapter 3 Fields A field is a set in the abstract algebra. The field of rational numbers ℚ, the field of real numbers ℝ and the field of complex number ℂ are more often seen in the fields. The field of complex number contains the field of real numbers, and the field of real numbers contains the field of rational numbers. So, the field of rational numbers ℚ is a subfield of the field of real numbers ℝ, and the field of real numbers ℝ is a subfield of the field of complex number ℂ, i.e. ℚ ⊆ ℝ⊆ ℂ. Definition 3.1.1 Field: Suppose a nonempty subset F of real numbers contains at least 2 elements and the set F is closed under two operations that are addition and multiplication. Then, the nonempty subset F is a field. Field is a set which addition, subtraction, multiplication, and division can be operated on non-zero elements of the set. The operation method is the same as the rational number and the real number. [3] For ∀ α, β ∈F, the operation as below: Addition: α + β= β+ α ∈F. Multiplication: α· β =β· α ∈F. And it is required that any element β in the set has an additive inverse element -β, and for all non-zero elements β has a multiplicative inverse element β-1. Then, we can define the inverse operation of addition and multiplication as below: α – β=α+(-β) ∈F. α/β=α· β-1 ∈F (where β≠0). Theorem 3.1.2. If the real numbers α and β are constructible numbers, then the number of α+β, α-β, αβ and α/β (where β≠0) are constructible. Under the operations of multiplication and addition, the set of all real numbers ℝ forms a field. So, we have the two corollaries as below: Corollary 3.1.3. A set of all constructible real numbers F is a subfield of the field of the real numbers ℝ. A set of all rational numbers is also a field. Corollary 3.1.4. A field of rational numbers ℚ is a subfield of the field of all constructible real numbers.

Chapter 4

The field extensions

4.1 The vector space Before discussing the field extensions, let me introduce another algebraic structure, the vector space. Some concepts of this algebraic structure have important applications in field extensions. Definition 4.1.1. Vector: A vector is a quantity which has two basic characteristics: magnitude and direction. The quantity corresponding to a vector is called a scalar, which has only magnitude but no direction. In physics, the line segment’s length represents the magnitude of a vector, and the arrow points represents the direction. In addition to being used to represent certain physical quantities, vectors can also perform mathematical operations, such as vector addition 푝⃗ + 푞⃗, and scalar multiplication α푝⃗ (where α is a scalar). Definition 4.1.2. Vector space: Suppose V is a non-empty set, the vector addition and scalar multiplication are defined on the non-empty set V. If the vector 푝⃗, 푞⃗ and 푟⃗ in the non-empty set V and each scalar α and β (where α, β∈F) are satisfied the following axioms, then V is a vector space over the field F. [4] 1. 푝⃗ + 푞⃗∈ V. 2. 푝⃗ + 푞⃗ = 푞⃗ + 푝⃗. 3. 푝⃗ + (푞⃗ + 푟⃗) = (푝⃗ + 푞⃗) + 푟⃗. 4. For all 푝⃗ in V, there exists a zero vector ⃗0⃗ in the non-empty set V. So, we have 푝⃗+⃗0⃗=푝⃗. 5. For all 푝⃗ in V, there exists a vector (-푝⃗) in the non-empty set V. So, we have 푝⃗ + (-푝⃗)= ⃗0⃗. 6. α푝⃗∈ V. 7. α (푝⃗ + 푞⃗) = α푝⃗ + α푞⃗. 8. (α+β) 푝⃗ = α푝⃗ + β푝⃗. 9. α (β푝⃗)= (αβ) 푝⃗. 10. 1푝⃗ = 푝⃗. Definition 4.1.3. Linear combination: Suppose V is a subset of the vector space over the field F. If there are a finite number of vectors p1, p2, …, pn ∈ V, and the corresponding scalar α1, α2, …, αn ∈ F such that V= α1p1+ α2p2 +…+ αnpn. Then the vector V is called a linear combination of the vectors p1, p2, …, pn.

Definition 4.1.4. Span: Suppose the vector space V over the field F, and P= {p1,

11 p2, …, pn} is a subset of V. All linear combinations of each vector in P is called the span of P, denoted by span(P).

Suppose P= {p1, p2, …, pn} is a vector set of vector space V over the field F, and the equation α1p1+ α2p2 +…+ αnpn = 0. If there exist α1, α2, …, αn ∈ F, where α1,

α2, …, αn are not all zero, then the vector set P is linearly dependent; Otherwise, the set P is linearly independent.

Definition 4.1.5. Basis: The vectors P= {p1, p2, …, pn} form a basis for a vector space V if and only if span(P)= V and the vector set P is linearly independent. A vector space always contains more than one basis, but each basis of a vector space contains the same number of basis vectors. We called the number be the dimension of a vector space. For example, since ℝ⊆ ℂ, the field of real number ℝ is a subfield of the field of complex number ℂ. The elements in the real number ℝ is closed under the operations of addition and multiplication. So, regardless of any two real numbers are added or multiplied, the result will be a real number. By the definition, suppose F and E are two fields. If F⊆E, then the field F is the subfield of the field E. So, the field E is considered to be a vector space over the field F. In the vector space, vector is the element which in the field E and scalar is the element which in the field F. As mentioned above, the field of real number ℝ is a subfield of the field of complex number ℂ. So, the field of complex number ℂ is a vector space over the field of real number ℝ, where the vector is a complex number, and the scalar is a real number. A basis for this vector space is {1, i}.

4.2 The field extensions

A field Extension is to add one or more elements that are not in the field F to the Field F to obtain a larger field, it is called the field extension. Some polynomial equations without solution in the field F, but we can find their roots in the field extension. So, the field extensions are related to finding the roots of polynomial equations. By the last section, the field of complex number ℂ is a field extension of the field of real numbers ℝ. For example, the equation x2+1 has no roots in the field of real number ℝ, since its roots are equal to ±√(−1), and the field of real numbers does not contain the square root of negative numbers. The equation has roots in the field of complex number ℂ, and its root is ±i. Definition 4.2.1. Field extension: Suppose F and E are two fields. If F⊆E, then the field E is the field extension of the field F, denoted by E/F. [5] If there exists a field H which satisfies F⊆H⊆E, then the field H is an intermediate field of E/F, and the field E is the field extension of the field H. Considering a field extension ℂ/ ℚ, we have the intermediate field ℚ(√훾)={α+β√훾|α, β∈ ℚ}. Definition 4.2.2 Simple extension: Given a field extension which is E/F, and there is E=F(α) for an element α∈ E. So that the field E is the simple extension of the field F. For example, ℚ(√훾) is the simple extension of the field of rational numbers ℚ, where 훾∈ ℤ+. Definition 4.2.3. Algebraic: Suppose E/F is a field extension. In the F(x), if there is a non-zero polynomial f (x) that satisfies f (α) = 0 (where α ∈ E), then α is algebraic over the field F. If not, it is called transcendental. Suppose F is a field and the field E is a finite extension of the field F, then the elements that in the field E are all algebraic over the field F. Definition 4.2.4. Degree of field extension: Suppose E/F is a field extension. So, the field E could be regarded as a vector space over the field F. The degree of the field extension, denoted by [E: F] which is the dimension of this vector space E. [6] If [E: F] is a finite integer, then the field extension E/F is a finite extension; Otherwise, it is an infinite extension. As I mentioned earlier about a basis for the vector space is {1, i} if the field of complex number ℂ is a vector space over the field of real number ℝ. Since the basis contains 2 elements, the dimension of the vector space is 2, i.e. [ℂ: ℝ] = 2. Definition 4.2.5. Minimal polynomial: Suppose E/F is a simple algebraic extension, where the leading polynomial p with the lowest degree rooted at α and its coefficient is 1. this p is uniquely determined. It is called the minimal polynomial of

α. For example, ℂ/ℝ is the simple algebraic extension, the minimal polynomial of i is x2+1. Considering the field extension which is ℚ(3√2)/ℚ, x3 – 2 is called the minimal polynomial of 3√2 over the field of rational numbers ℚ. It is monic and irreducible with 3√2 as a root. Since the coefficient of x3 is 1. Since x3 – 2 could not be factored into a product of polynomials of lower degree in ℚ[x], it is prime in ℚ[x]. The extension field ℚ(3√2) is a vector space over the field of rational numbers ℚ, and the basis is {1, 3√2, (3√2)2}. So, the dimension of this vector space is 3, i.e. [ℚ(3√2) : ℚ] = 3.

4.3 The algebraic extensions

Definition 4.3.1. Algebraic extension: If the elements in an extensional field of F are all algebraic over the field F, we called that this extension be an algebraic extension. If we add an element (√3) into the field ℚ(√2), then we have the field extension ℚ(√2) (√3)/ ℚ(√2), and the simple extension is ℚ(√2,√3)/ ℚ. For the field extension which is ℚ(√2)/ ℚ, since the coefficient of x2 is 1, x2-2 is the minimal polynomial of √2. Since x2 – 2 cannot be factored into a product of polynomials of lower degree in ℚ[x], it is prime in ℚ[x]. So, it is monic and irreducible with √2 as a root. The extension field ℚ(√2) is a vector space over the field of rational numbers ℚ, and the basis is {1, √2}. So, the dimension of this vector space is 2, i.e. [ℚ(√2) : ℚ] = 2. Similarly, the extension field ℚ(√2) (√3) is a vector space over the field ℚ(√2), and its basis is equal to {1, √3}. Thus, the dimension of this vector space is 2, i.e. [ℚ(√2) (√3) : ℚ(√2)] = 2. So, the basis of ℚ(√2,√3)/ ℚ is equal to {1,√2, √3, √6}, and the dimension of this vector space is 4, i.e. [ℚ(√2,√3) : ℚ] = 4. Theorem 4.3.2. Suppose E/F is the field extension, the field H is the intermediate field of the field extension E/F, then E/F is a finite extension if and only if both E/H and H/F are finite extension. We have [E : F]=[E : H] [H : F]. [7] Proof: Since E/F is the field extension and the field H is the intermediate field of the field extension E/F, then F⊆H⊆E. The field E is a field extension of the field H.

So, the field E is a vector space over the field H, and the vectors S= {u1, u2, …, um} form a basic for the vector space E. Let p∈ the field E. We have:

p = α1u1+ α2u2 +…+ αmum, where αj ∈ H and u1, u2, …, um ∈S Since F⊆H, the field H is a field extension of the field F. So, the field H is a vector space over the field F, and the vectors T= {v1, v2, …, vn} form a basic for the vector space H, we have

αj=β1,jv1+ β2,jv2+ … + βn-1,jvn-1+ βn,jvn, where βi,j∈ F So,

p= (β1,1v1+ β2,1v2+ … + βn,1vn)p1+ … +(β1,mv1+ β2,mv2+ … +βn,mvn)pm

=β1,1v1p1+ … +βi,jvi pj+ … +βn,mvnpm If p=0. i.e.

(β1,1v1+ β2,1v2+ … + βn,1vn)p1+ … +(β1,mv1+ β2,mv2+ … +βn,mvn)pm=0 By the linearly independent theory, we have

β1,jv1+ β2,jv2+ … + βn-1,jvn-1+ βn,jvn=0 and βi,j=0.

All of the αj must be 0 since the {pj}are linearly independent, and then all of the βi,j

15 must be 0 since the {vi} are linearly independent. So, {vipj} are linearly independent.

It means that the {vipj} are a basis. So, [E : F]=[E : H] [H : F].

Chapter 5 Doubling the Cube Suppose the length of the original cube is 1 unit, and the length of the cube to be constructed is x. To prove that it is impossible to double the cube, it is necessary to satisfy that the equation x3=2 has no rational roots. So, by the straightedge and compass construction, there does not exist any ratio number x can be drawn. Theorem 5.1.1. If α is a constructible number, then α is algebraic over the field of rational numbers ℚ. So, a power of 2 is the degree of the minimal polynomial in ℚ[x] of α. Proof. A constructible number refers the real number that can be constructed in a limit number of steps by using a straightedge. In the case of a given unit length, if we can use a straightedge to construct a line segment of length |α| (where α∈ ℝ), then the number α is constructible. The field of rational numbers ℚ is a subfield of the field of all constructible real numbers. By section 2.1, we know that we can obtain √α (where α>0) by taking a square root under the four mathematical operation: addition, subtraction, multiplication and division. So, √α is constructible.

Let ℚn= ℚ(√α1,√α2, √α3, … , √αn), where √αi ∈ ℕ.

For the simple extension of the field of rational numbers ℚ, ℚ(√α1)/ ℚ. The minimal 2 polynomial of √α1 is x -α1. The extension field ℚ(√α1) is a vector space over the field of rational numbers ℚ, and the basis is {1, √α1}. If α1=0, then [ℚ(√α1) : ℚ]=1;

If α1>0, then [ℚ(√α1) : ℚ]=2. Thus, [ℚ(√α1) : ℚ]=1 or 2.

Since ℚ⊆ ℚ(√α1) ⊆ ℚ(√α1,√α2) ⊆ … ⊆ ℚ(√α1,√α2, √α3, … , √αn), we have

[ℚ(√α1,√α2, √α3, … , √αn) : ℚ]=[ ℚ(√α1) : ℚ][ ℚ(√α1,√α2) :

ℚ(√α1)]…[ℚ(√α1,√α2, √α3, … , √αn) : ℚ(√α1,√α2, √α3, … , √αn-1)]. m m Each product equals 1 or 2. So, [ℚ(√α1,√α2, √α3, … , √αn) : ℚ] =2 . i.e. [ℚn: ℚ] =2 , where 0≤m≤n. m So, 2 =[ℚn: ℚ]= [ ℚn : ℚ(α)] [ℚ(α) : ℚ].

By the section 4.3, we know that [ℚ(√2) : ℚ]=2. If we add an element √√2 into the field ℚ(√2), then we have [ℚ(√√2) ∶ ℚ]=[ ℚ(√√2) ∶ ℚ(√2)][ ℚ(√2) : ℚ]=

4=22. Theorem 5.1.2 It is impossible to double the cube by using the straightedge and compass construction. [8] Proof. Suppose the length of the original cube is 1 unit, and the length of the cube to be constructed is x. We have the equation x3=2. By the section 4.2, we know

17 that the extension field ℚ(3√2) is a vector space over the field of rational numbers ℚ, its basis is {1, 3√2, (3√2)2}. So, the dimension of this vector space is 3, i.e. [ℚ(3√2) : ℚ] = 3. By the theorem 5.1.1, if 3√2 is a constructible number, there exists a solution for the equation 2m=3, where m∈ ℤ+. However, such m does not exist. It means that 3√2 cannot construct by the straightedge and compass construction. So, it is impossible to double the cube.

Chapter 6 Conclusion

To conclude, it is impossible to double the cube as 3√2 cannot construct by the straightedge and compass construction. The first proof of Theorem 5.1.1 and 5.1.2 is completed by Pierre Wantzel. The problem of doubling a cube is one of three geometrical construction problems, but the proof which is impossible to duplicate the cube was completed by algebraic method. To prove that it is possible to double the cube, it is necessary to satisfy that the equation x3=2 has rational roots. The extension field ℚ(3√2) is a vector space over the field of rational numbers ℚ, and the basis is {1, 3√2, (3√2)2}. So, the dimension of this vector space is equal to 3, i.e. [ℚ(3√2) : ℚ] = 3. But there is not exists a solution for the equation 2m=3, where m∈ ℤ+. So, 3√2 is not a constructible number. It means that 3√2 cannot construct by the straightedge and compass construction. So, it is impossible to duplicate the cube.

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