Geometric Constructible Numbers

In summary, the conversation is discussing geometric constructible numbers and the fact that if 'a' is constructible, then its algebraic order is 2^n. However, the inverse statement is not necessarily true. The individual is looking for counterexamples to this statement and has not found a clear answer yet.
  • #1
emptyboat
28
1
Geometric Constructible Numbers...

Hi, everyone.
I have a question about geometric constructible numbers.
I know that "if 'a' is constructible then [Q(a):Q]=2^n."
But I heard that its inverse is not true.
I want some counter examples about the inverse statement.
(I have checked by googling 'i' is a constructible number.)

Help...
 
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  • #2


You mean you want a number that is algebraic of order a power of 2 and is NOT a constructible number? Hmm, now thats' a good question! I will need to think about that- for a few decades.
 
  • #3


Yes, I mean that.
So, there was no clear answer about that?
 
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What are geometric constructible numbers?

Geometric constructible numbers are numbers that can be constructed using only a straightedge and compass. This means that they can be created by drawing straight lines and circles, and using basic geometric constructions such as intersecting lines and bisecting angles.

What is the significance of geometric constructible numbers?

Geometric constructible numbers have been studied for centuries and have played an important role in mathematics. They were first studied by the ancient Greeks and have been used in various fields, such as geometry, algebra, and number theory. They also have connections to other mathematical concepts such as root extraction and field extensions.

How are geometric constructible numbers different from algebraic numbers?

Geometric constructible numbers are a subset of algebraic numbers, which are any numbers that can be expressed as the solution of a polynomial equation with rational coefficients. While all geometric constructible numbers are algebraic, not all algebraic numbers are geometric constructible. This is because geometric constructions can only create numbers that are the solution to quadratic equations, while algebraic numbers can also be the solution to higher-degree polynomial equations.

What are some examples of geometric constructible numbers?

Some examples of geometric constructible numbers include the square root of 2, the golden ratio, and the cube root of 2. These numbers can be constructed by using a straightedge and compass, as well as other basic geometric constructions.

How are geometric constructible numbers related to Euclidean geometry?

Geometric constructible numbers are closely related to Euclidean geometry, as they are based on the principles of Euclidean constructions. Euclidean geometry is the study of geometric figures and shapes using only a straightedge and compass, and geometric constructible numbers are numbers that can be created using these same tools.

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