Proof of the impossibility of trisecting the angle

Builds on Constructible number Trisecting the angle de Moivre's formula Pythagorean trigonometric identity Rational root theorem Polynomials of degree two or three are irreducible iff they have no roots Minimal polynomial of an algebraic element The minimal polynomial divides every polynomial vanishing at the element Degree of a simple extension is the degree of the minimal polynomial The degree of a constructible number is a power of two The constructible numbers form a field closed under square roots Euler's formula Complex conjugate Modulus of a complex number A complex number times its conjugate

Proof

Let $\theta = \pi/9$ , let $c = \cos\theta$ and $s = \sin\theta$ , and let $\mathcal{C}$ be the set of the constructible numbers.

By de Moivre's formula, $\cos 3\theta + i \sin 3\theta = (c + i s)^3 = c^3 + 3 i c^2 s - 3 c s^2 - i s^3$ .
Comparing real parts, and using $s^2 = 1 - c^2$ , we get $\cos 3\theta = c^3 - 3c(1 - c^2) = 4c^3 - 3c$ .
Since $\cos 3\theta = \cos(\pi/3) = 1/2$ , we get $8c^3 - 6c - 1 = 0$ . So $u = 2c$ is a root of $g = x^3 - 3x - 1$ .
By the rational root theorem, a rational root of $g$ is $\pm 1$ . But $g(1) = -3$ and $g(-1) = 1$ . So $g$ has no rational root.
A cubic with no root in $\Q$ is irreducible over $\Q$ . So the minimal polynomial of $u$ , which divides the monic $g$ , equals $g$ .
By the degree of a simple extension, $[\Q(u) : \Q] = 3$ . Since $c = u/2$ and $u = 2c$ , we have $\Q(c) = \Q(u)$ , so $[\Q(c) : \Q] = 3$ .
The degree of a constructible number is a power of $2$ , and $3$ is not. So $\cos(\pi/9) \notin \mathcal{C}$ .

Now we turn to the construction problem. The angle $\pi/3$ has vertex $0$ and sides through $1$ and $\omega = e^{i\pi/3}$ . A trisection would construct a point $w \neq 0$ on the ray from $0$ at angle $\pi/9$ .

By the field theorem, $\mathcal{C}$ is a subfield of $\C$ closed under square roots, and it is the smallest such subfield.
By Euler's formula, $\omega = 1/2 + i\sqrt{3}/2$ . Here $i$ is a square root of $-1$ and $\sqrt 3$ a square root of $3$ , so $\omega \in \mathcal{C}$ . So the angle $\pi/3$ can be drawn.
Let $\overline{\mathcal{C}}$ be the set of the conjugates of the elements of $\mathcal{C}$ . Conjugation preserves the field operations, so $\overline{\mathcal{C}}$ is a subfield of $\C$ .
If $w' = \bar z$ with $z \in \mathcal{C}$ and $t^2 = w'$ , then $\bar t^2 = z$ , so $\bar t \in \mathcal{C}$ and $t \in \overline{\mathcal{C}}$ . So $\overline{\mathcal{C}}$ is closed under square roots.
By minimality, $\mathcal{C} \subseteq \overline{\mathcal{C}}$ . So each $z \in \mathcal{C}$ is $\bar u$ for some $u \in \mathcal{C}$ , and then $\bar z = u \in \mathcal{C}$ .
Suppose that the point $w$ is constructed. The given points $0$ , $1$ and $\omega$ lie in $\mathcal{C}$ , so $w \in \mathcal{C}$ . Then $\bar w \in \mathcal{C}$ , so $w \bar w = \abs{w}^2$ is in $\mathcal{C}$ . Its square root, the modulus $\abs{w}$ , is in $\mathcal{C}$ .
Since $w$ lies on the ray at angle $\theta$ , we have $w = \abs{w} e^{i\theta}$ , so $e^{i\theta} = w / \abs{w} \in \mathcal{C}$ .
Then $\cos\theta = (e^{i\theta} + \overline{e^{i\theta}})/2 \in \mathcal{C}$ , by Euler's formula and step 12. This contradicts step 7.
So the angle $\pi/3$ cannot be trisected with straightedge and compass, and $\cos(\pi/9)$ is not constructible.

Idea

The triple angle formula makes $2\cos(\pi/9)$ a root of the irreducible cubic $x^3 - 3x - 1$ . So its degree over $\Q$ is $3$ , which is not a power of $2$ . A trisection would produce the point $e^{i\pi/9}$ , and then its real part would be constructible.

Reading path

To understand this page from the ground up, read these 474 pages in order. The foundations come first. The path takes the proofs Proof of the Cauchy–Hadamard theorem, Proof of Bézout's identity, Proof that a function with nonnegative second derivative is convex, Proof of the derivative of cosine by term-by-term differentiation, Proof of the Weierstrass M-test, Proof of de Moivre's formula from Euler's formula, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of cancellation in an integral domain, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the formula for the derivative of a polynomial, Proof of the generalized Euclid's lemma, Proof of the rational root theorem, Proof that polynomials of degree two or three are irreducible iff they have no roots, Proof of the commutativity of addition by induction, Proof of term-by-term differentiation, Proof of the universal property of the polynomial ring, Proof of the epsilon-delta form of continuity, Proof of the power rule by induction, Proof that absolutely convergent series converge, Proof of the sum of a finite geometric series, Proof of the description of the nth roots of unity, Proof that convergent sequences are Cauchy, Proof that a complex number times its conjugate is the square of its modulus, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule with an auxiliary difference quotient, Proof that the constructible numbers form the smallest subfield of the complex numbers closed under square roots, Proof of the Cauchy criterion for series, Proof of the tower law, Proof that exp(2 pi i/n) is a primitive root of unity, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof of Euclid's lemma, Proof that the exponential series has infinite radius of convergence, Proof that R[x] is an integral domain iff R is, Proof that cos pi = -1 and sin pi = 0, Proof of the equality of complex numbers, Proof that a function with nonnegative derivative is increasing, Proof that finite extensions are algebraic, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the root test, Proof that the degree of a simple extension is the degree of the minimal polynomial, Proof of the double negation law by a truth table, Proof that the logarithm is the inverse of the exponential, Proof that power series converge uniformly on smaller intervals inside the radius of convergence, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof of the first isomorphism theorem for rings, Proof that a ring homomorphism is injective iff its kernel is zero, Proof that the exponential function is positive, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the uniqueness of limits, Proof of the degree of a product over an integral domain, Proof of strong induction from ordinary induction, Proof that there are exactly n nth roots of unity, Proof that power series are differentiable inside the radius of convergence, Proof of the limit of a product of sequences, Proof that uniform convergence of derivatives allows term-by-term differentiation, Proof of the correspondence theorem for rings, Proof that the degree of a constructible number is a power of two, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the existence of square roots of nonnegative reals, Proof of Pascal's rule from the recursive definition, Proof that a differentiable function is convex iff its derivative is increasing, Proof that i squared is minus one, Proof of the linearity of differentiation, Proof of the angle addition formulas by differentiation, Proof that R/M is a field iff M is maximal, Proof of the sequential characterization of limits, Proof that the exponential function is strictly increasing, Proof that Cauchy sequences are bounded, Proof that the only ideals of a field are zero and the whole field, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the existence of nth roots of complex numbers, Proof that the modulus is multiplicative, Proof of the comparison test, Proof of the existence of the infimum by negation, Proof of the remainder theorem, Proof of the continuity of the inverse of a strictly monotone continuous function, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof that every linearly independent set extends to a basis, by Zorn's lemma, Proof that K(a) is isomorphic to K[x] modulo the minimal polynomial, Proof that 1 + x is at most exp x, Proof of the factorial formula for binomial coefficients by induction, Proof that a function with positive derivative is strictly increasing, Proof that the modulus of the conjugate equals the modulus, Proof of Zorn's lemma, Proof of the product of exponentials by the Cauchy product, Proof of the quadratic formula by completing the square, Proof that the complex numbers form a field, Proof of the intermediate value theorem by the supremum, Proof of the binomial theorem by induction, Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms, Proof that differentiable functions are continuous, Proof that one-sided limits of a monotone function exist, Proof that sine and cosine are 2π-periodic with smallest period 2π, Proof of the epsilon-delta form of the limit of a function, Proof of the derivative of sine by term-by-term differentiation, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof that the minimal polynomial is irreducible, Proof of the division algorithm, Proof of the Archimedean property in an Archimedean ordered field, Proof of the existence and uniqueness of the principal value of the argument, Proof that the minimal polynomial divides every polynomial vanishing at the element, Proof of the polar form of a complex number, Proof that a composition of continuous functions is continuous, Proof of the exponential of a sum by the Cauchy product, Proof of the Pythagorean trigonometric identity by differentiation, Proof that a decreasing sequence bounded below converges, Proof of the term test, Proof that the exponential function is its own derivative, by term-by-term differentiation, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that convergent sequences are bounded, Proof that K[x]/(f) is a field iff f is irreducible, Proof of the product rule, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that in a PID a nonzero element is irreducible iff it generates a maximal ideal, Proof that a function with zero derivative on an open interval is constant, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof of the factor theorem, Proof of the ratio formula for the radius of convergence, Proof of Euler's formula by power series, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the mean value theorem from Rolle's theorem, Proof that the Cauchy product of absolutely convergent series converges absolutely to the product, Proof that slopes of a convex function are increasing, Proof that every vector space has a basis, by Zorn's lemma, Proof that Euclidean domains are principal ideal domains, Proof of the universal property of the quotient ring, Proof that the exponential function is strictly convex, Proof that K[x] is a Euclidean domain, Proof of the constant multiple rule.

Binomial coefficient definition
Complex numbers object
Complex numbers as pairs of reals concept
Construction of the reals from Cauchy sequences concept
Convex function definition
Definition by recursion concept
Divisibility definition
Element definition
Even and odd functions definition
Factorial definition
Group concept
Imaginary unit object
Integers object
Limit point of a set definition
Linear combination concept
Local maximum definition
Mathematical induction concept
Natural numbers object
Ordered field definition
Peano axioms concept
Periodic function definition
Proof of Zorn's lemma proof
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Spanning set concept
Straightedge and compass construction concept
Strictly monotone function definition
Vector space definition
Proof of Pascal's rule from the recursive definition proof
Complex exponential definition
Power series in the complex plane definition
Real and imaginary parts definition
Root of unity definition
Proof that slopes of a convex function are increasing proof
Strictly convex function definition
Greatest common divisor definition
Prime number definition
Proof that i squared is minus one proof
Set definition
Ring definition
Ring of integers statement
Limit of a function definition
Local minimum definition
Archimedean ordered field definition
Successor function definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Constructible number definition
Trisecting the angle problem
Linear combination definition
Linear subspace definition
Pascal's rule statement
Complex conjugate definition
Primitive root of unity definition
Slopes of a convex function are increasing statement
Coprime integers definition
i squared is minus one statement
Family of sets definition
Ordered pair definition
Ideal definition
Commutative ring definition
Polynomial ring definition
Ring homomorphism definition
Subring definition
Unit of a ring definition
Zero divisor definition
Union definition
Addition of natural numbers definition
Continuity at a point definition
Derivative definition
One-sided limit definition
Proof of the Archimedean property in an Archimedean ordered field proof
The rationals form a field statement
The successor function is injective statement
Zero is not a successor statement
Biconditional definition
Truth table concept
Modus ponens concept
Existential quantifier definition
Universal quantifier definition
Bounded sequence definition
Cauchy sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Sequence tending to infinity definition
Series definition
Completeness axiom statement
Linear independence definition
Linear span definition
Proof of the factorial formula for binomial coefficients by induction proof
Proof of the equality of complex numbers proof
Cartesian product definition
Maximal ideal definition
Two-sided ideal definition
Ideal generated by a subset definition
Divisibility in a commutative ring definition
Degree of a polynomial definition
Polynomial evaluation definition
Proof of the universal property of the polynomial ring proof
Field definition
Kernel of a ring homomorphism definition
Ring isomorphism definition
Integral domain definition
Proof of the commutativity of addition by induction proof
Subset definition
Union of a family of sets definition
Order on the natural numbers definition
Continuous function definition
Archimedean property statement
Differentiable function definition
One-sided derivative definition
The rationals form an ordered field statement
Propositional logic concept
Mathematical proof concept
Uniqueness quantifier definition
Limit superior definition
Proof of the sum of a finite geometric series proof
Convergent sequence definition
Pointwise convergence definition
Sequential continuity definition
The sequence r^n tends to infinity for r > 1 example
Cauchy product definition
Partial sum definition
Power series definition
Spanning set definition
Equality of complex numbers statement
Factorial formula for binomial coefficients statement
Principal ideal definition
Leading coefficient definition
Quotient ring definition
Root of a polynomial definition
Binary relation definition
Proof that the only ideals of a field are zero and the whole field proof
Evaluation homomorphism definition
Proof of cancellation in an integral domain proof
Subfield definition
Order on the reals definition
Euclidean domain definition
Irreducible element definition
Commutativity of addition statement
Proof that a composition of continuous functions is continuous proof
Proof of strong induction from ordinary induction proof
Derivative function definition
Tautology definition
Sum of a finite geometric series statement
Uniform convergence definition
Radius of convergence definition
Convergent series definition
Sine and cosine definition
Finite-dimensional vector space definition
Basis definition
Proof of the binomial theorem by induction proof
Monic polynomial definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Proof that R/M is a field iff M is maximal proof
Principal ideal domain definition
Proof of the universal property of the quotient ring proof
Proof of the degree of a product over an integral domain proof
The only ideals of a field are zero and the whole field statement
Proof of the remainder theorem proof
Proof that the complex numbers form a field proof
Universal property of the polynomial ring statement
Field extension definition
Cancellation in an integral domain statement
Interval definition
Irreducible polynomial definition
Function definition
Strong induction statement
Higher derivative definition
Uniform convergence on a set definition
Logical equivalence definition
Dimension definition
Dimension theorem for vector spaces statement
Sum of a series definition
Series of functions definition
Absolute convergence definition
Binomial theorem statement
Preorder definition
Division algorithm for polynomials statement
Degree of a product statement
Proof that in a PID a nonzero element is irreducible iff it generates a maximal ideal proof
Universal property of the quotient ring statement
Algebraic element definition
The complex numbers form a field statement
Intermediate field definition
Degree of a field extension definition
Proof of the epsilon-delta form of the limit of a function proof
Image of a set definition
Preimage of a set definition
Proof of the epsilon-delta form of continuity proof
Proof of the well-ordering principle from strong induction proof
Choice function definition
Codomain definition
Injective function definition
Second derivative definition
Proof of the double negation law by a truth table proof
Partial order definition
Proof of the quadratic formula by completing the square proof
Minimal polynomial of an algebraic element definition
In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
Remainder theorem statement
Proof that R[x] is an integral domain iff R is proof
Finite extension definition
Monotone function definition
Algebraic extension definition
Field generated by adjoining elements definition
Epsilon-delta definition of a limit statement
Proof that a ring homomorphism is injective iff its kernel is zero proof
Epsilon-delta definition of continuity statement
Well-ordering principle statement
Axiom of choice statement
Surjective function definition
Double negation law statement
Maximal element definition
Total order definition
Upper bound definition
Proof that the minimal polynomial divides every polynomial vanishing at the element proof
Proof of the factor theorem proof
Proof that Euclidean domains are principal ideal domains proof
R[x] is an integral domain iff R is statement
Proof that the minimal polynomial is irreducible proof
Simple extension definition
A ring homomorphism is injective iff its kernel is zero statement
Proof of the division algorithm proof
Composition of continuous functions is continuous statement
Bijective function definition
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Chain definition
The minimal polynomial divides every polynomial vanishing at the element statement
Factor theorem statement
Proof that K[x] is a Euclidean domain proof
Euclidean domains are principal ideal domains statement
Division algorithm statement
The minimal polynomial is irreducible statement
Proof of the first isomorphism theorem for rings proof
Proof of the correspondence theorem for rings proof
Equinumerous sets definition
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
K[x] is a Euclidean domain statement
Correspondence theorem for rings statement
Zorn's lemma statement
Proof of Bézout's identity proof
First isomorphism theorem for rings statement
Proof that polynomials of degree two or three are irreducible iff they have no roots proof
Cardinality definition
The reals form an ordered field statement
Bézout's identity statement
R/M is a field iff M is maximal statement
Polynomials of degree two or three are irreducible iff they have no roots statement
Absolute value definition
Proof that every linearly independent set extends to a basis, by Zorn's lemma proof
Proof that every vector space has a basis, by Zorn's lemma proof
Proof of the approximation property of the supremum proof
Proof of the existence of square roots of nonnegative reals proof
Proof of the existence of the infimum by negation proof
Proof of the generalized Euclid's lemma proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof that K[x]/(f) is a field iff f is irreducible proof
Every vector space has a basis statement
Proof of the epsilon-N definition of the limit of a sequence proof
Existence of square roots of nonnegative reals statement
Every linearly independent set extends to a basis statement
Approximation property of the supremum statement
Existence of the infimum statement
Triangle inequality statement
Absolute value is multiplicative statement
Generalized Euclid's lemma statement
K[x]/(f) is a field iff f is irreducible statement
Square root definition
Proof that finite extensions are algebraic proof
Proof of the tower law proof
Epsilon-N definition of the limit of a sequence statement
Proof that one-sided limits of a monotone function exist proof
Proof that the reals are Archimedean from the completeness axiom proof
Proof of the intermediate value theorem by the supremum proof
Modulus of a complex number definition
Proof of the limit of a sum of sequences proof
Proof of the squeeze theorem for sequences proof
Proof of Euclid's lemma proof
Finite extensions are algebraic statement
Proof of the uniqueness of limits proof
Proof that K(a) is isomorphic to K[x] modulo the minimal polynomial proof
Proof that limits preserve non-strict inequalities proof
Tower law statement
Proof that differentiable functions are continuous proof
Intermediate value theorem statement
Proof that subsequences of a convergent sequence converge to the same limit proof
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
The reals are Archimedean statement
The sequence 1/n tends to zero example
Proof that Cauchy sequences are bounded proof
Proof of the monotone convergence theorem for sequences proof
Proof of the term test proof
Argument of a complex number definition
Proof that the modulus of the conjugate equals the modulus proof
Proof that a complex number times its conjugate is the square of its modulus proof
K(a) is isomorphic to K[x] modulo the minimal polynomial statement
Euclid's lemma statement
Floor function definition
Proof that there is a natural number n with 1/n below any positive epsilon proof
Limit of a sum of sequences statement
Uniqueness of limits statement
Differentiable functions are continuous statement
Limits preserve non-strict inequalities statement
Proof of the continuity of the inverse of a strictly monotone continuous function proof
Proof of the ratio formula for the radius of convergence proof
Squeeze theorem statement
Subsequences of a convergent sequence converge to the same limit statement
Convergent sequences are bounded statement
Convergent sequences are Cauchy statement
Cauchy sequences are bounded statement
Monotone convergence theorem for sequences statement
Modulus of the conjugate statement
A complex number times its conjugate statement
Proof of the rational root theorem proof
Proof of the limit of a product of sequences proof
There is a natural number n with 1/n below any positive epsilon statement
Proof that the degree of a simple extension is the degree of the minimal polynomial proof
One-sided limits of a monotone function exist statement
Term test statement
Proof that a decreasing sequence bounded below converges proof
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
The sequence r^n tends to zero for |r| < 1 example
Proof that the modulus is multiplicative proof
Rational root theorem statement
Degree of a simple extension is the degree of the minimal polynomial statement
Decreasing sequence bounded below converges statement
Proof of the sequential characterization of limits proof
Continuity of the inverse of a strictly monotone continuous function statement
Limit of a product of sequences statement
Proof that continuity is equivalent to sequential continuity proof
A series of nonnegative terms converges iff its partial sums are bounded statement
Geometric series example
Modulus is multiplicative statement
Proof of Euler's formula by power series proof
Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
Sequential characterization of limits statement
Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms proof
Continuity is equivalent to sequential continuity statement
Proof of the Cauchy–Hadamard theorem proof
Proof of the root test proof
Proof of the ratio test proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Proof of the comparison test proof
Euler's formula statement
Comparison test statement
Proof of the product rule proof
Proof of the linearity of differentiation proof
Proof of the chain rule with an auxiliary difference quotient proof
Bolzano–Weierstrass theorem statement
Cauchy product of absolutely convergent series statement
Exponentials dominate powers in sequences statement
Proof of the product of exponentials by the Cauchy product proof
Linearity of differentiation statement
Product rule statement
Chain rule statement
Ratio test statement
Root test statement
Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Product of exponentials statement
Proof of the constant multiple rule proof
Boundedness theorem statement
Ratio formula for the radius of convergence statement
Proof of the power rule by induction proof
Cauchy–Hadamard theorem statement
Cauchy criterion for sequences statement
Constant multiple rule statement
Proof of de Moivre's formula from Euler's formula proof
Power rule statement
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Uniformly Cauchy sequence of functions definition
Proof of the Cauchy criterion for series proof
de Moivre's formula statement
Extreme value theorem statement
Proof of Fermat's theorem on stationary points proof
Proof of the formula for the derivative of a polynomial proof
Cauchy criterion for series statement
Derivative of a polynomial statement
Proof that absolutely convergent series converge proof
Fermat's theorem on stationary points statement
Proof of Rolle's theorem proof
Absolutely convergent series converge statement
Rolle's theorem statement
Proof of the Weierstrass M-test proof
Weierstrass M-test statement
Proof of the mean value theorem from Rolle's theorem proof
Mean value theorem statement
Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof
Proof that a function with nonnegative derivative is increasing proof
Proof that a function with zero derivative on an open interval is constant proof
Proof that a differentiable function is convex iff its derivative is increasing proof
Proof that uniform convergence of derivatives allows term-by-term differentiation proof
Power series converge uniformly on smaller discs inside the radius of convergence statement
Proof that a function with positive derivative is strictly increasing proof
A function with nonnegative derivative is increasing statement
A function with zero derivative is constant statement
A differentiable function is convex iff its derivative is increasing statement
Uniform convergence of derivatives allows term-by-term differentiation statement
A function with positive derivative is strictly increasing statement
Proof that the exponential series has infinite radius of convergence proof
Proof that a function with nonnegative second derivative is convex proof
The exponential series has infinite radius of convergence statement
Proof of term-by-term differentiation proof
A function with nonnegative second derivative is convex statement
Term-by-term differentiation statement
Exponential function definition
Proof of the exponential of a sum by the Cauchy product proof
Proof that power series are differentiable inside the radius of convergence proof
Exponential of a sum statement
Power series are differentiable inside the radius of convergence statement
Proof that the exponential function is positive proof
Proof that the exponential function is its own derivative, by term-by-term differentiation proof
Proof of the derivative of cosine by term-by-term differentiation proof
Proof of the derivative of sine by term-by-term differentiation proof
The exponential function is positive statement
The exponential function is its own derivative statement
Derivative of cosine statement
Derivative of sine statement
Proof that the exponential function is strictly increasing proof
Proof of the angle addition formulas by differentiation proof
Proof of the Pythagorean trigonometric identity by differentiation proof
pi object
The exponential function is strictly increasing statement
Pythagorean trigonometric identity statement
Angle addition formulas statement
Arccosine definition
Proof that 1 + x is at most exp x proof
Proof that the exponential function is strictly convex proof
Natural logarithm definition
Proof that cos pi = -1 and sin pi = 0 proof
The exponential function is convex statement
cos pi = -1 statement
1 + x is at most exp x statement
Proof that sine and cosine are 2π-periodic with smallest period 2π proof
Proof that the logarithm is the inverse of the exponential proof
Sine is 2π-periodic statement
Logarithm is the inverse of the exponential statement
Proof of the existence and uniqueness of the principal value of the argument proof
Principal value of the argument statement
Proof that exp(2 pi i/n) is a primitive root of unity proof
Proof of the polar form of a complex number proof
Polar form statement
Proof of the existence of nth roots of complex numbers proof
Proof of the description of the nth roots of unity proof
The nth roots of unity statement
exp(2 pi i/n) is a primitive root of unity statement
Proof that there are exactly n nth roots of unity proof
Number of nth roots of unity statement
Existence of nth roots of complex numbers statement
Quadratic formula statement
Proof that the constructible numbers form the smallest subfield of the complex numbers closed under square roots proof
The constructible numbers form a field closed under square roots statement
Proof that the degree of a constructible number is a power of two proof
The degree of a constructible number is a power of two statement
Proof of the impossibility of trisecting the angle proof

Where this leads

2 pages build on this one.

Tag	`QB6RCHAYK0`
URI	`m2://QB6RCHAYK0`
Kind	proof
Writing	W0 stub
MSC	12F05
Proves	Impossibility of trisecting the angle
Uses	Constructible number, Trisecting the angle, de Moivre's formula, Pythagorean trigonometric identity, Rational root theorem, Polynomials of degree two or three are irreducible iff they have no roots, Minimal polynomial of an algebraic element, The minimal polynomial divides every polynomial vanishing at the element, Degree of a simple extension is the degree of the minimal polynomial, The degree of a constructible number is a power of two, The constructible numbers form a field closed under square roots, Euler's formula, Complex conjugate, Modulus of a complex number, A complex number times its conjugate
Changed	4 November 2026

Proof of the impossibility of trisecting the angle

Proof

Idea

Reading path

Where this leads

Further reading

What this page needs for W1

What links here