Undergrad "Easy example" of transcendental numbers

[Hill]

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:

How should I notice that x is transcendental?

 

[jedishrfu]

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:

$(x^2-2)=0$

 

[fresh_42]

$x$ 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 $p(x)=0.$

 

[Hill]

Why

there is no polynomial $p(t)\in \mathbb{Q}[t]$ such that $p(x)=0$

?

 

[fresh_42]

Why

?

What is $x$?

 

[Hill]

What is $x$?

I don't know. This is my question.

 

[fresh_42]

I read it as an indeterminate and as such does not fulfill any equation, except $x\cdot 0=0.$

 

[Hill]

So, where is an example of a transcendental number in this?

 

[fresh_42]

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.

 

[martinbn]

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.