# 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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### Real number known not to be a period

I am working a bit with problems in non-archimedean settings inspired by the famous periods conjecture by Kontsevich-Zagier. I was preparing a talk and wanted to give of background about the initial ...
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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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### Regarding Algebraic Dependence

Let $\Omega \supset F$ be a field extension and $A,B\subset \Omega$ be two sets. Def $1$: $A$ is algebraically independent over $F$ if every finite subset of $A$ is algebraically independent over $F$...
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Simple Question: Is the proof given below correct? Let $t$ satisfy $2^t+3^t=1$. We have $t \approx -0.787884911025869783628555917298434738269083137354182194199 \dots$ according to wolfram alpha. Then $... • 3,650 3 votes 1 answer 104 views ### Simple estimation of separation of powers of 2 and powers of 3? An accepted answer is in the cross post at https://mathoverflow.net/questions/428396/simple-estimation-of-difference-of-powers-of-2-and-powers-of-3 . 1. Question How to get from the formulas $$\left| ... • 438 6 votes 1 answer 178 views ### Problem with Veblen's proof for the transcendence of \pi I'm trying to understand the following proof, (no need to read all of it), but there's one point where I'm stuck. The proof is by contradiction, so they use the definition of an algebraic number (a is ... • 63 0 votes 0 answers 25 views ### k\subset K field extension, S transcendence base, then K is algebraic extension of k(S), why? Terminology from Lang, Chapter VIII. 1. k\subset K field extension and S transcendence base, then K is algebraic over k(S). Why is that so? Lang states as it is something trivial. Surely we ... • 167 4 votes 1 answer 140 views ### How to prove that k^{\pi} is not an integer for any integer k\geq 2? I strongly suspect that k^{\pi} is not an integer for any integer k\geq 2 (for otherwise this would be a famous result of which I am not aware). But how does one prove this? The answer to this ... • 15.8k 0 votes 2 answers 81 views ### Prove that \{x\} is transcendental base of \mathbb{Q}(x,i) We have the field extension \mathbb{C}/\mathbb{Q}. Let x\in \mathbb{R}\subset \mathbb{C} be transcendental over \mathbb{Q}. Show, that \{x\} is a transcendental basis of L = \mathbb{Q}(x,i). ... • 1,574 0 votes 0 answers 29 views ### Question about purely transcendental extension Let F be a field of characteristic \neq 2 and let u be transcendental over F. Suppose u^2+v^2=1. Show that F(u,v) is a purely transcendental extension by showing that F(u,v) = F(\frac{1+v}... • 512 3 votes 0 answers 33 views ### Is there any p \in \mathbb{Q}[x,y] with x and y degree both at least 1 such that p(\pi,e) is known to be irrational? The algebraic independence of \pi and e is a well known open problem, as is the specific case of rationality of \pi + e. My question is if there is any polynomial with rational coefficients (... 2 votes 0 answers 60 views ### Transcendence of Liouville-Like Numbers Liouville numbers such as$$\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 ... 2 votes 1 answer 303 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 ... • 5,041 2 votes 1 answer 108 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... 0 votes 0 answers 17 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 ... 1 vote 1 answer 43 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... 1 vote 1 answer 63 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? 1 vote 0 answers 91 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? 0 votes 1 answer 42 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. ... • 2,969 0 votes 1 answer 86 views ### 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, ... • 1,937 3 votes 0 answers 66 views ### 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 ... • 1,310 3 votes 1 answer 56 views ### 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 ... • 79.8k -2 votes 1 answer 64 views ### 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 ... 8 votes 0 answers 197 views ### 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}!}$$where F_{n} is the n^{th} Fibonacci number. Is it possible to prove that it will converge to a transcendental number? Edit: Proof of ... • 81 4 votes 1 answer 143 views ### Transcendence of \pi+\log\alpha and e^{\alpha\pi+\beta} In the book "Transcendental Number Theory" by Alan Baker, he proves a few corollaries of Baker's theorem. I've attached this page below. After, he claims that special cases of these ... • 100 1 vote 1 answer 87 views ### Explanation of proof of transcendence of e I'm following the proof for the transcendence of e from the book "Transcendental Numbers" by M. Ram Murty and Purusottam Rath. I am struggling to understand the final few lines. As far as ... • 100 1 vote 1 answer 46 views ### Explanation of proof of the transendence of \alpha_1^{\beta_1} \dotsm \alpha_n^{\beta_n} I'm following this proof in the book "Transcendental Numbers" by M. Ram Murty and Purusottam Rath. The result is a corollary of Baker's theorem. There are a couple of things I don't ... • 100 3 votes 1 answer 76 views ### What makes Hilbert's 7th problem important/relevant? [closed] What was the motivation behind Hilbert's 7th problem? Looking into some of the history behind transcendental number theory, it seems that the field was almost non-existent in the late 1800's/early ... 0 votes 2 answers 52 views ### Span of Transcendental Numbers Let a and b be two transcendental numbers. Does there exist r \in \mathbb{R} such that r cannot be expressed as any finite (integral) powers of a and b with rational coefficients? For any ... 0 votes 1 answer 36 views ### Proving equivalence between a summation expression and a power expression I have that$$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}^{\... 43 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. ...
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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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### What methods show that a number is transcendental?

I've been doing a lot of research on such theories lately and these are all I've found so far: Liouvilles criterion (here) Lindemann-Weierstrass theorem (here) Gelfond-Schneider theorem (here) ...
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### Liouville-Roth Irrationality Measure of $\pi$ = 2 already proven?

I was looking through this paper and I was curious. Has it already been established that the Flint Hills series $\displaystyle\sum_{n\in\mathbb{Z}^{+}}\frac{\csc^2 n}{n^3}$ converges? And has it ...
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### Is the number transcendental?

Consider the following number: $R=\frac{1}{9}\sum^\infty_{n=1} 10^{-\frac{n\left(n+1\right)}{2}}\left(10^n-1\right)\left(n\left(\operatorname{mod}10\right)\right)$ =0....
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### Rigidity of logarithms of positive integers

The following question is from my colleague. It seems to be emerged considering some elementary number theory problem. This is NOT as exercise or problem in published material although it may looks ...
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### Flat $\mathbb{Z}$-modules and algebraic independence
I know that this question is naive, but I really want to find out if this observation is correct. First, let $\alpha_1, \alpha_2, \dotsc, \alpha_k$ be real numbers. Consider the $\mathbb{Z}$-module ...
Let's note that where $\{x\}$ means fractional part of $x$. I am trying to figure out if $\{e^n\}_{n \in \mathbb{N}}$ and $\{\pi^n\}_{n \in \mathbb{N}}$ are dense in $[0,1]$. In general do we know if ...