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?
kingwinner said:Is it a+b*pi, a,b E Q?
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.
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.
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.
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.
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.