# Smallest number field containing pi

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!

## Answers and Replies

Dick
Science Advisor
Homework Helper
First show the field must be infinite. Then the smallest it can be is countable. To show it's countable try to construct it.

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?

Last edited:
Dick
Science Advisor
Homework Helper
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?

Is it a+b*pi, a,b E Q?

Dick
Science Advisor
Homework Helper
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.