# 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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### Prove that $\{x\}$ is transcendental base of $\mathbb{Q}(x,i)$

We have the field extension $\mathbb{C}/\mathbb{Q}$. Let $x\in \mathbb{R}\subset \mathbb{C}$ be transcendental over $\mathbb{Q}$. Show, that $\{x\}$ is a transcendental basis of $L = \mathbb{Q}(x,i)$. ...
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Let $F$ be a field of characteristic $\neq 2$ and let $u$ be transcendental over $F$. Suppose $u^2+v^2=1$. Show that $F(u,v)$ is a purely transcendental extension by showing that $F(u,v) = F(\frac{1+v}... 0answers 28 views ### Is there any$p \in \mathbb{Q}[x,y]$with x and y degree both at least 1 such that$p(\pi,e)$is known to be irrational? The algebraic independence of$\pi$and$e$is a well known open problem, as is the specific case of rationality of$\pi + e$. My question is if there is any polynomial with rational coefficients (... 0answers 38 views ### Transcendence of Liouville-Like Numbers Liouville numbers such as $$\sum_{k=1}^\infty\frac1{10^{k!}}$$ are known to be transcendental, essentially from Diophantine approximation type arguments. Using stronger results than what Louiville had ... 1answer 131 views ### Books for transcendental number theory I would like to start reading about transcendental numbers. I am familiar with the basics of field theory, number fields, and complex analysis. I have the least exposure to Galois theory. I am ... 1answer 100 views ### Field automorphism of$F(t)$Let$t$be transcendental over$F$. Let$a,b,c,d \in F$such that$ad-bc \ne 0$. Prove that there is a field automorphism of$F(t)$given by$\sigma (t) = \frac{at+b}{ct+d}$and$\sigma(\alpha)=\alpha$... 0answers 16 views ### A question about transcendental extensions Consider an extension$K/F$. Suppose$S \subset K$such that$S$is algebraically independent over$F$. Prove that each$s \in S$is transcendental over$F(S-\{s\})$. How do I prove this claim? Any ... 1answer 35 views ###$t+1$transcendent over$K$if$t$is transcendent I've been solving some problems from my Galois Theory course and I want to check if the solution I came up with is correct. The question was: Given that an element$t$is transcendent over a field$K$... 1answer 30 views ### How to prove that any rational linear combination of u and v will also be transcendent? Let u a algebraic number and v a transcendent number, so any rational linear combination of u and v will also be transcendent. How to prove it? 0answers 80 views ### If$e^{ei\pi}$turns out to be transcendent, can it be proved that$e^e$is also? [duplicate] We know that$e^e$is not proved to be transcendent, so neither is the number$e^{ei\pi}$. If$e^{ei\pi}$turns out to be transcendent, can it be proved that$e^e$is also? 1answer 33 views ### How can I prove that I can find$N$real elements that is algebraically independent over$\mathbb{Q}$for any N$?

I was wondering how can one show that there exist $N$ real elements that is algebraically independent over $\mathbb{Q}$ for any $N$? (I was thinking perhaps Lindemann–Weierstrass theorem can be used. ...
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### Reference Request: Gelfond Schneider Theorem

I want to learn the background for the Gelfond Schneider Theorem: https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem I want to learn the background needed for the proof of this theorem, ...
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### Field Extensions with Common Transcendental Numbers

Does anyone know if there is work done in this direction where one extends (the field) $\mathbb{Q}$ or $\bar{\mathbb{Q}}$ with certain common transcendental numbers such as $\pi$, $e$, etc. For ...
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### Non-zero coefficient in a transcendence proof

I am studying the proof of the simple version of Lindemann's Theorem. Theorem. If $\alpha$ is a non-zero algebraic number, then $e^\alpha$ is transcendental. Apologies for the long read, but to ...
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### How can I prove this question?

We know that it is not proved that $e^e$ is transcendental, so neither is the number that $e^{e\sqrt{2}}$. My question is, if one turns out to be, how can it be proved that the other is? Because there ...
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### Is the sum of the inverses of the factorials of Fibonacci numbers transcendental?

This is the sum, $$e'=\sum_{n=1}^\infty \frac{1}{F_{n}!}$$ where $F_{n}$ is the $n^{th}$ Fibonacci number. Is it possible to prove that it will converge to a transcendental number? Edit: Proof of ...
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### Transcendence of $\pi+\log\alpha$ and $e^{\alpha\pi+\beta}$

In the book "Transcendental Number Theory" by Alan Baker, he proves a few corollaries of Baker's theorem. I've attached this page below. After, he claims that special cases of these ...
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### Explanation of proof of transcendence of $e$

I'm following the proof for the transcendence of $e$ from the book "Transcendental Numbers" by M. Ram Murty and Purusottam Rath. I am struggling to understand the final few lines. As far as ...
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### Explanation of proof of the transendence of $\alpha_1^{\beta_1} \dotsm \alpha_n^{\beta_n}$

I'm following this proof in the book "Transcendental Numbers" by M. Ram Murty and Purusottam Rath. The result is a corollary of Baker's theorem. There are a couple of things I don't ...
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### What makes Hilbert's 7th problem important/relevant? [closed]

What was the motivation behind Hilbert's 7th problem? Looking into some of the history behind transcendental number theory, it seems that the field was almost non-existent in the late 1800's/early ...
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### Span of Transcendental Numbers

Let $a$ and $b$ be two transcendental numbers. Does there exist $r \in \mathbb{R}$ such that $r$ cannot be expressed as any finite (integral) powers of $a$ and $b$ with rational coefficients? For any ...
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I have that $$2^{-(a+1)!}(1+\frac{1}{2^{1}}+\frac{1}{2^{2}}+...)=2(2^{-a!})^{a+1}\,\,\,\,\,\,\,\,(1)$$ Which I am trying to show is $\geq \sum_{b=a+1}^{\infty}2^{-b!}$ in order to prove $\sum_{b=0}^{\... 1answer 41 views ### 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. ... 0answers 28 views ### 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$(x,f(x))$... 0answers 46 views ### 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 ... 1answer 39 views ### 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).$0answers 31 views ###$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$-... 0answers 37 views ### Clarification on the solution set of two surfaces being a hyperbola$xy=e$is a hyperbola. Looking at the LHS and RHS we have a hyperbolic parabaloid (a conoid) and a constant. Taken together, the equation yields a hyperbola. Manipulating algebraically, gives$xy=e^{\...
From Vladimir Zorich Analysis I: Let us call an irrational number $a \in \mathbb{R}$ well approximated by rational numbers if for any natural number $n, N \in \mathbb{N}$ there exists a rational ...