Proving C is the Smallest Constructible Numbers Subfield with a>0 Property

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In summary, Constructible Numbers are numbers that can be obtained using a straightedge and compass, and are limited to the operations of addition, subtraction, multiplication, and division. These numbers are unique as they can only be obtained through construction, and have played a significant role in the history of mathematics, leading to the discovery of new concepts and techniques. While not directly applicable in the real world, the study of Constructible Numbers has influenced fields such as engineering, architecture, and computer graphics.
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Homework Statement



Show that the field C of constructible numbers is the smallest subfield of R with the property that [tex]a\in C, a>0 \Rightarrow \sqrt{a}\in C[/tex].

The Attempt at a Solution



Suppose there's a proper subfield of C' of C that has that property, then let [tex]a\in C-C'[/tex]. Somehow I must show that a is actually in C. Perhaps repeated squaring?
 
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Maybe a different tactic would be useful?

Let C' be the smallest subfield with that property. Can you prove C is a subfield of C', and that C has that property?
 

1. What are Constructible Numbers?

Constructible Numbers are numbers that can be obtained using a straightedge and compass. This means that these numbers can be constructed geometrically, without the use of any other tools or measurements.

2. How are Constructible Numbers different from other numbers?

Constructible Numbers are unique because they can only be obtained by construction, while other numbers can be calculated or expressed in various ways. Additionally, Constructible Numbers are limited to the operations of addition, subtraction, multiplication, and division, while other numbers may have more complex operations defined for them.

3. Can any number be constructed using a straightedge and compass?

No, not all numbers are constructible. For example, pi and the square root of 2 are not constructible, as they require more complex operations such as square roots. Only numbers that can be expressed as a finite combination of square roots, addition, subtraction, multiplication, and division of whole numbers are considered constructible.

4. What is the significance of Constructible Numbers in mathematics?

Constructible Numbers have played an important role in the history of mathematics and have been studied by many prominent mathematicians. They are closely related to other mathematical concepts such as geometry, algebra, and number theory. Additionally, the study of Constructible Numbers has led to the discovery of new mathematical concepts and the development of new mathematical techniques.

5. How are Constructible Numbers used in real-world applications?

While Constructible Numbers may not have direct applications in the real world, the concepts and techniques used to study them have been applied in fields such as engineering, architecture, and computer graphics. The study of Constructible Numbers also has implications for the understanding of symmetry and patterns in nature.

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