The degree of a constructible number is a power of two

Builds on The constructible numbers form a field closed under square roots Tower law Degree of a simple extension is the degree of the minimal polynomial Minimal polynomial of an algebraic element

Statement

Let $z \in \mathbb{C}$ be a constructible number. Then $z$ is algebraic over $\mathbb{Q}$ , and $[\mathbb{Q}(z) : \mathbb{Q}] = 2^r$ for some integer $r \ge 0$ .

Remarks

The proof: there is a tower $\mathbb{Q} = F_0 \subseteq F_1 \subseteq \dots \subseteq F_n$ with $z \in F_n$ and $[F_j : F_{j-1}] = 2$ for each $j$ . By the tower law, $[F_n : \mathbb{Q}] = 2^n$ , and $[\mathbb{Q}(z) : \mathbb{Q}]$ divides it. So the minimal polynomial of $z$ over $\mathbb{Q}$ has degree a power of two. The converse is false: some algebraic numbers whose degree is a power of two are not constructible. A sufficient condition: the splitting field of the minimal polynomial of $z$ has degree a power of two over $\mathbb{Q}$ .

Reading path

To understand this page from the ground up, read these 56 pages in order. The foundations come first.

Group concept
Straightedge and compass construction concept
Ring definition
Constructible number definition
Commutative ring definition
Ideal definition
Polynomial ring definition
Ring homomorphism definition
Subring definition
Unit of a ring definition
Zero divisor definition
Divisibility in a commutative ring definition
Ideal generated by a subset definition
Maximal ideal definition
Two-sided ideal definition
Degree of a polynomial definition
Polynomial evaluation definition
Field definition
Integral domain definition
Kernel of a ring homomorphism definition
Ring isomorphism definition
Principal ideal definition
Quotient ring definition
The only ideals of a field are zero and the whole field statement
Evaluation homomorphism definition
Leading coefficient definition
Root of a polynomial definition
Subfield definition
Irreducible element definition
Euclidean domain definition
Correspondence theorem for rings statement
Principal ideal domain definition
The constructible numbers form a field closed under square roots statement
Monic polynomial definition
Universal property of the quotient ring statement
Irreducible polynomial definition
Degree of a product statement
Field extension definition
R/M is a field iff M is maximal statement
In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
Division algorithm for polynomials statement
Algebraic element definition
R[x] is an integral domain iff R is statement
First isomorphism theorem for rings statement
Intermediate field definition
Degree of a field extension definition
Minimal polynomial of an algebraic element definition
K[x] is a Euclidean domain statement
Field generated by adjoining elements definition
Tower law statement
The minimal polynomial divides every polynomial vanishing at the element statement
K[x]/(f) is a field iff f is irreducible statement
Simple extension definition
K(a) is isomorphic to K[x] modulo the minimal polynomial statement
Degree of a simple extension is the degree of the minimal polynomial statement
The degree of a constructible number is a power of two statement

Where this leads

5 pages build on this one.

Tag	`0SSSM1D475`
URI	`m2://0SSSM1D475`
Kind	statement, theorem
Writing	W0 stub
MSC	12F05
Status	This copy of the site has no build data. Run `m2 status 0SSSM1D475`.
Uses	The constructible numbers form a field closed under square roots, Tower law, Degree of a simple extension is the degree of the minimal polynomial, Minimal polynomial of an algebraic element

The degree of a constructible number is a power of two

Statement

Remarks

Reading path

Where this leads

Further reading

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