# Questions tagged [transcendence-theory]

A brief study of transcendental numbers and algebraic independence theories; currently an on-development branch of mathematics with a lot of open problems.

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### Quantity of Numbers of different irrationality-levels [closed]

Preconditions: a mathematical irrational number $x$ is given to be irrational if it can not be expressed by a fraction $\frac{a}{b}$ where $a,b\in \mathbb{Z}$. from now on let's call the normal ...
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### Images of a vector under the Galois differential group span the solution set

I am reading the paper "A refined version of the Siegel-Shidlovskii theorem" by F. Beukers. In the proof of Theorem 1.5, he mentions the following results in Galois differential theory. Let ...
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### Surjective maps between algebraic groups induce surjective maps between connected components

I am reading the paper of F. Beukers, A refined version of the Siegel-Shidlovskii theorem (here is the link). The author mentions the following result in Algebraic group theory without proof. Lemma 2....
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### Does Schanuel's conjecture imply that $\pi^e$ is transcendental?

My understanding (and correct me if I'm wrong) is that it is unknown whether $\pi^e$ is algebraic or transcendental. I've also been led to believe that most open questions of this type would be solved ...
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### Transcendence measure for the canonical Liouville number

Let $\displaystyle\alpha=\sum_{n=0}^{+\infty}\frac1{10^{n!}}$. It is well-known that $\alpha$ is transcendental. I am looking for a transcendental measure for $\alpha$. That is exercise 11.15 of the ...
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### Show, using a specific approach, that $\dim \Bbb P^n=\dim\Bbb A^n=n$.

This is Exercise 1.8.4(1) of Springer's, "Linear Algebraic Groups (Second Edition)". It is not a duplicate of The dimension of $\mathbb P^n$ is $n$ because I'm after a particular perspective;...
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### Prove $2^{\frac{1}{\log_2(x)}}$ is algebraic iff $x=2^n$ or $1/2^n$ for some $n \in \Bbb Q^+.$

Conjecture: $2^{\frac{1}{\log_2(x)}}$ is algebraic iff $x=2^n$ or $1/2^n$ for some $n\in \Bbb Q^+.$ How can I prove my conjecture? It might be very easy to prove but I am stuck at the moment. ...
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### Do any ordered pairs of algebraic numbers satisfy this function?

Q: Does there exist an algebraic number $x,$ s.t. $f(x)$ is also an algebraic number? $f(x)=\exp\bigg(\frac{1}{\ln(x)}\bigg)$ for $x\ne0,1.$ I would like to prove that the set of points $(x,f(x))$ ...
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### Field extensions equal

Let $F$ be a field and $F(x_1,x_2)$ be a finite seperable extension. Let $t_1,t_2$ be algebraically independent over $F$. Let $u=t_1x_1+t_2x_2$. Prove that $F(t_1,t_2,u)=F(t_1,t_2,x_1,x_2)$. One ...
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### Prove that $e^{\frac{1}{\log(x)}}$ is at least countably transcendental

Q: Prove that $f(x)=e^{\frac{1}{\log(x)}}$ is at least countably transcendental for $x\in\Bbb R\cap (0,1).$
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### $F$-embedding is an automorphism if and only if ${\rm tr. deg}(E/F)<\infty$.

Let $E/F$ be a field extension with $E$ algebraically closed. Show that every $F$-embedding $E \to E$ is an automorphism if and only if ${\rm tr. deg}(E/F ) < \infty$. Sufficiency: We have a $F$-...
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### What does algebraic independence mean?

If you search for algebraic independence, you will find the following on Wikipedia: In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S ...
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