Proof that the degree of a constructible number is a power of two

Builds on Constructible number The constructible numbers form a field closed under square roots Subfield Field generated by adjoining elements 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 Tower law Finite extensions are algebraic

Proof

Call a finite chain of subfields $\Q = F_0 \subseteq F_1 \subseteq \dots \subseteq F_m$ of $\C$ a square-root tower if, for each $j$ , $F_j = F_{j-1}(s_j)$ for some $s_j \in \C$ with $s_j^2 \in F_{j-1}$ . The case $m = 0$ is allowed.
Let $T$ be the set of the complex numbers that lie in the top field $F_m$ of some square-root tower.
In a square-root tower, $s_j$ is a root of $x^2 - s_j^2 \in F_{j-1}[x]$ . So its minimal polynomial over $F_{j-1}$ divides this polynomial and has degree $1$ or $2$ .
By the degree of a simple extension, $[F_j : F_{j-1}] \in \set{1, 2}$ . By the tower law, $[F_m : \Q]$ is a product of $1$ s and $2$ s, so it is a power of $2$ .
Let $z, w \in T$ , with towers $(F_j)_{j \le m}$ built from $s_1, \dots, s_m$ and $(G_j)_{j \le l}$ built from $t_1, \dots, t_l$ .
Extend the first tower by $F_{m+j} = F_{m+j-1}(t_j)$ for $1 \le j \le l$ . By induction on $j$ , $G_j \subseteq F_{m+j}$ : indeed $G_j = G_{j-1}(t_j)$ , and $F_{m+j}$ contains $G_{j-1}$ and $t_j$ .
So $t_j^2 \in G_{j-1} \subseteq F_{m+j-1}$ , and the extended chain is a square-root tower. Its top field contains $z$ and $w$ .
That top field is a field. So $z + w$ , $-z$ , $zw$ , and $z^{-1}$ for $z \neq 0$ , lie in it and hence in $T$ . Also $\Q \subseteq T$ , from the tower with $m = 0$ . So $T$ is a subfield of $\C$ .
Let $z \in T$ lie in the top field $F_m$ of a tower, and let $s \in \C$ with $s^2 = z$ . Then $F_{m+1} = F_m(s)$ extends the tower, so $s \in T$ . So $T$ is closed under square roots.
The constructible numbers form the smallest subfield of $\C$ closed under square roots. By steps 8 and 9, every constructible number lies in $T$ .
Let $z$ be constructible. By step 10, $z \in F_m$ for some square-root tower, and $[F_m : \Q] = 2^e$ for some $e \ge 0$ by step 4.
So $F_m / \Q$ is finite, hence algebraic, and $z$ is algebraic over $\Q$ .
Since $\Q \subseteq \Q(z) \subseteq F_m$ , the tower law gives $2^e = [F_m : \Q(z)] \cdot [\Q(z) : \Q]$ . So $[\Q(z) : \Q]$ divides $2^e$ .
By unique factorisation, a positive divisor of $2^e$ is $2^r$ for some $r$ with $0 \le r \le e$ . So $[\Q(z) : \Q] = 2^r$ .

Idea

The numbers reachable from $\Q$ by adjoining square roots one at a time form a field closed under square roots, so they contain all constructible numbers. Each adjunction at most doubles the degree. So a constructible number sits in a field of degree $2^e$ , and its own degree divides $2^e$ .

Reading path

To understand this page from the ground up, read these 454 pages in order. The foundations come first. The path takes the proofs Proof of the Cauchy–Hadamard theorem, 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 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 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 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 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 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
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
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
i squared is minus one statement
Family of sets definition
Ordered pair definition
Commutative ring definition
Polynomial ring definition
Ring homomorphism definition
Subring definition
Unit of a ring definition
Zero divisor definition
Ideal definition
Union definition
Continuity at a point definition
Derivative definition
One-sided limit definition
Addition of natural numbers definition
Proof of the Archimedean property in an Archimedean ordered field proof
Biconditional definition
The rationals form a field statement
The successor function is injective statement
Truth table concept
Zero is not a successor statement
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
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
Ideal generated by a subset definition
Maximal ideal definition
Two-sided ideal definition
Subset definition
Continuous function definition
Union of a family of sets definition
Archimedean property statement
Differentiable function definition
Proof of the commutativity of addition by induction proof
One-sided derivative definition
Order on the natural numbers definition
Propositional logic concept
The rationals form an ordered field statement
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
Leading coefficient definition
Root of a polynomial definition
Evaluation homomorphism definition
Binary relation definition
Subfield definition
Proof of cancellation in an integral domain proof
Euclidean domain definition
Proof that the only ideals of a field are zero and the whole field proof
Irreducible element definition
Principal ideal definition
Quotient ring definition
Order on the reals definition
Proof that a composition of continuous functions is continuous proof
Derivative function definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
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
Monic polynomial definition
Proof of the binomial theorem by induction proof
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Proof of the degree of a product over an integral domain proof
Field extension definition
Universal property of the polynomial ring statement
Proof that the complex numbers form a field proof
Cancellation in an integral domain statement
Irreducible polynomial definition
The only ideals of a field are zero and the whole field statement
Proof of the universal property of the quotient ring proof
Proof that R/M is a field iff M is maximal proof
Principal ideal domain definition
Interval definition
Function definition
Higher derivative definition
Strong induction statement
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
Algebraic element definition
Degree of a field extension definition
Intermediate field definition
The complex numbers form a field statement
Universal property of the quotient ring statement
Proof that in a PID a nonzero element is irreducible iff it generates a maximal ideal proof
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
Choice function definition
Codomain definition
Injective function definition
Second derivative definition
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Minimal polynomial of an algebraic element definition
Proof of the quadratic formula by completing the square proof
Algebraic extension definition
Finite extension definition
Field generated by adjoining elements definition
Proof that R[x] is an integral domain iff R is proof
Monotone function definition
Epsilon-delta definition of a limit statement
Proof that a ring homomorphism is injective iff its kernel is zero proof
In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
Epsilon-delta definition of continuity statement
Axiom of choice statement
Surjective function definition
Well-ordering principle statement
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
Simple extension definition
Proof that the minimal polynomial is irreducible proof
R[x] is an integral domain iff R is statement
A ring homomorphism is injective iff its kernel is zero statement
Proof that Euclidean domains are principal ideal domains proof
Composition of continuous functions is continuous statement
Proof of the division algorithm proof
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
Proof that K[x] is a Euclidean domain proof
The minimal polynomial is irreducible statement
Division algorithm statement
Proof of the first isomorphism theorem for rings proof
Euclidean domains are principal ideal domains statement
Equinumerous sets definition
Proof of the correspondence theorem for rings proof
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
Cardinality definition
Zorn's lemma statement
First isomorphism theorem for rings statement
Correspondence theorem for rings statement
The reals form an ordered field statement
Absolute value definition
R/M is a field iff M is maximal statement
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 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
Approximation property of the supremum statement
Every linearly independent set extends to a basis statement
Existence of the infimum statement
Triangle inequality statement
Absolute value is multiplicative 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
Finite extensions are algebraic statement
Proof that K(a) is isomorphic to K[x] modulo the minimal polynomial proof
Tower law statement
Proof of the uniqueness of limits proof
Proof that limits preserve non-strict inequalities proof
Proof that differentiable functions are continuous proof
Intermediate value theorem statement
Proof that subsequences of a convergent sequence converge to the same limit proof
The reals are Archimedean statement
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
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
Proof that there is a natural number n with 1/n below any positive epsilon proof
Floor function definition
Limit of a sum of sequences statement
Differentiable functions are continuous statement
Uniqueness of limits 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 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
Degree of a simple extension is the degree of the minimal polynomial statement
Proof of the sequential characterization of limits proof
Decreasing sequence bounded below converges statement
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
Continuity is equivalent to sequential continuity statement
Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms proof
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
Proof of the power rule by induction proof
Ratio formula for the radius of convergence statement
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 zero derivative on an open interval is constant proof
Proof that a differentiable function is convex iff its derivative is increasing proof
Proof that a function with nonnegative 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 zero derivative is constant statement
A function with nonnegative derivative is increasing 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

Where this leads

10 pages build on this one.

Tag	`DVQR23MRT7`
URI	`m2://DVQR23MRT7`
Kind	proof
Writing	W0 stub
MSC	12F05
Proves	The degree of a constructible number is a power of two
Uses	Constructible number, The constructible numbers form a field closed under square roots, Subfield, Field generated by adjoining elements, 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, Tower law, Finite extensions are algebraic
Changed	4 November 2026

Proof that the degree of a constructible number is a power of two

Proof

Idea

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