Smallest number field containing pi

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In summary, the conversation discusses how to prove that the smallest number field containing pi is countable. It is necessary to first show that the field is infinite and then construct it to prove that it is countable. The field must contain Q and pi, and it can also contain other elements such as a+b*pi where a and b are rational numbers. It is important to be able to perform basic operations like addition, subtraction, and multiplication within the field.
  • #1
kingwinner
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1) Let F be the smallest number field containing pi. Prove that F is countable.

The first trouble I am facing is: What actually is the smallest number field containing pi? How can we know its cardnality?

Could someone please help me? Thanks!
 
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  • #2
First show the field must be infinite. Then the smallest it can be is countable. To show it's countable try to construct it.
 
  • #3
But what is the field?

e.g. the smallest number field is the set of rational numbers

What is the smallest number field containing pi? How can we find the cardnality without knowing what it is?
 
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  • #4
kingwinner said:
But what is the field?

e.g. the smallest number field is the set of rational numbers

What is the smallest number field containing pi? How can we find the cardnality without knowing what it is?

I told you. YOU have to construct it. You can't look it up in a table of fields. It has to contain Q and it has to contain pi. What else does it have to contain give that it contains Q and pi?
 
  • #5
Is it a+b*pi, a,b E Q?
 
  • #6
kingwinner said:
Is it a+b*pi, a,b E Q?

That's a good start. Now you can add and subtract. But you have to be able to multiply in the field as well. So it had better contain pi^2, right? Better keep expanding the field.
 

FAQ: Smallest number field containing pi

What is the smallest number field containing pi?

The smallest number field containing pi is the field of algebraic numbers, which consists of all numbers that can be obtained by combining rational numbers and square roots of rational numbers using the four basic arithmetic operations.

How is pi related to number fields?

Pi is related to number fields because it is a transcendental number, meaning it cannot be expressed as a root of any polynomial equation with rational coefficients. This makes pi an element of the field of algebraic numbers, which is the smallest number field containing pi.

Can pi be a part of other number fields?

No, pi can only be a part of the field of algebraic numbers. This is because all other number fields contain only algebraic numbers, which are numbers that can be expressed as roots of polynomial equations with rational coefficients. Since pi is transcendental, it cannot be expressed in this way and therefore cannot be a part of any other number field.

What are some properties of the field of algebraic numbers containing pi?

The field of algebraic numbers containing pi is an infinite field, meaning it contains infinitely many elements. It is also a commutative field, meaning the order in which operations are performed does not affect the result. Additionally, it is a subfield of the field of complex numbers, as all elements in the field of algebraic numbers can be expressed as complex numbers with a zero imaginary part.

Why is it important to study the smallest number field containing pi?

Studying the smallest number field containing pi can provide insights into the properties of pi and other transcendental numbers. It also has applications in number theory and algebraic geometry. Additionally, understanding the structure of this field can help us better understand and solve mathematical problems that involve transcendental numbers.

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