"Easy example" of transcendental numbers

  • #1
Hill
728
573
I am watching this lecture by Richard E Borcherds on Galois theory: Field extensions and the following "easy example" is not easy for me to understand.

Here it is screen-by-screen:
Screenshot 2024-12-13 140036.png

Screenshot 2024-12-13 140121.png

Screenshot 2024-12-13 140151.png

Screenshot 2024-12-13 140219.png

Screenshot 2024-12-13 140238.png

Screenshot 2024-12-13 140352.png

Screenshot 2024-12-13 140441.png

Screenshot 2024-12-13 140503.png


How should I notice that x is transcendental?
 

Attachments

  • Screenshot 2024-12-13 140036.png
    Screenshot 2024-12-13 140036.png
    19.5 KB · Views: 1
  • Screenshot 2024-12-13 140310.png
    Screenshot 2024-12-13 140310.png
    23.5 KB · Views: 2
Physics news on Phys.org
  • #2
Transcendental numbers are defined as irrational but not being in any other subset of irrationals.

Trancendental numbers are irrational with non-repeating, and non-terminating decimals but not algebraic like the ##\sqrt{2}## ie they can't be expressed as the solution to:

##(x^2-2)=0##
 
Last edited:
  • #3
##x## is transcendental if it is not algebraic. Algebraic numbers are those which are a root of a rational polynomial, and there is no polynomial ##p(t)\in \mathbb{Q}[t]## such that ##p(x)=0.##
 
  • #4
Why
fresh_42 said:
there is no polynomial ##p(t)\in \mathbb{Q}[t]## such that ##p(x)=0##
?
 
  • #5
Hill said:
Why

?
What is ##x##?
 
  • #6
fresh_42 said:
What is ##x##?
I don't know. This is my question.
 
  • #7
I read it as an indeterminate and as such does not fulfill any equation, except ##x\cdot 0=0.##
 
  • #8
So, where is an example of a transcendental number in this?
 
  • #9
Hill said:
So, where is an example of a transcendental number in this?
If you consider ##\mathbb{Q}(x)## the field of rational functions, i.e. the quotient of polynomials in ##x.## It is the same field as if we adjointed ##\pi##
$$
\mathbb{Q}(x) \cong \mathbb{Q}(\pi).
$$
The only difference is, that we need Lindemann's proof to see that ##\pi## is transcendental whereas an indeterminate is transcendental per definition of an indeterminate as a variable that does not fulfill an algebraic equation.
 
  • Like
Likes Hill
  • #10
Hill said:
So, where is an example of a transcendental number in this?
Read carefully, he doesn't say transcendetal number. He says that ##x## is transcendental.
 
  • Like
Likes Hill
Back
Top