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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A transcendental function?
Let $(u_n)_n=(p_n/q_n)_n$ $\left(\text{with $\mathrm{GCD}(p_n,q_n)=1$ and }\ln|u_n|=o(2^n)\right)$ a sequence of rational numbers that is not ultimately zero.
I want to prove that the function $f(z)=\...
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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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Let $L = F(S)$ for $S \subseteq L$. Does there exists a transcendental basis $T \subseteq S$ with $F(S) = F(T)$?
Let $L/F$ be a field extension with $L = F(S)$ for a subset $S \subseteq L$. My first question is:
Is there a subset $T \subseteq S$ such that $T$ is a transcendental basis?
Now let's say we have $I ...
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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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Is the statement of Baker's Theorem on wikipedia inaccurate?
The Wikipedia article on Baker's theorem states it as follows:
If $\lambda_1,\ldots,\lambda_n\in\mathbb{L}$ are linearly independent over the rational numbers, then for any algebraic numbers $\beta_0,...
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Are $\pi$ and $e^{\pi i \sqrt{d}}$ algebraically independent?
Let $-d\in\mathbb{N}$ be square-free. Nesterenko (https://doi.org/10.1070/SM1996v187n09ABEH000158) proved that $e^{\pi\sqrt{-d}}$ and $\pi$ are algebraically independent. Is it known whether $e^{\pi\...
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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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Schanuel's Conjecture $\implies $ no surprises on the integers. $2^t+3^t=1 \implies t\notin\overline{\mathbb{Q}}$
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 $...
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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| ...
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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 ...
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$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 ...
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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 ...
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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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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}...
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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 (...
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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 ...
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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 ...
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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$...
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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 ...
user910011
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$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$...
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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?
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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?
user892441
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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 ...
user892441
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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}!}$$ where $F_{n}$ is the $n^{th}$ Fibonacci number.
Is it possible to prove that it will converge to a transcendental number?
Edit: Proof of ...
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Transcendence of $\pi+\log\alpha$ and $e^{\alpha\pi+\beta}$
In the book "Transcendental Number Theory" by Alan Baker, he proves a few corollaries of Baker's theorem. I've attached this page below.
After, he claims that special cases of these ...
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Explanation of proof of transcendence of $e$
I'm following the proof for the transcendence of $e$ from the book "Transcendental Numbers" by M. Ram Murty and Purusottam Rath. I am struggling to understand the final few lines.
As far as ...
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Explanation of proof of the transendence of $\alpha_1^{\beta_1} \dotsm \alpha_n^{\beta_n}$
I'm following this proof in the book "Transcendental Numbers" by M. Ram Murty and Purusottam Rath.
The result is a corollary of Baker's theorem.
There are a couple of things I don't ...
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What makes Hilbert's 7th problem important/relevant? [closed]
What was the motivation behind Hilbert's 7th problem?
Looking into some of the history behind transcendental number theory, it seems that the field was almost non-existent in the late 1800's/early ...
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Span of Transcendental Numbers
Let $a$ and $b$ be two transcendental numbers.
Does there exist $r \in \mathbb{R}$ such that $r$ cannot be expressed as any finite (integral) powers of $a$ and $b$ with rational coefficients?
For any ...
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Proving equivalence between a summation expression and a power expression
I have that $$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}^{\...
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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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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^{\...
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Showing that Liouville's numbers are transcendental (Liouville's theorem)
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 ...
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What would be some implications of Schanuel's conjecture being proven wrong?
Schanuel's conjecture is an important conjecture in transcendental number theory, which is:
Given any $n$ complex numbers $z_1, z_2, ..., z_n$ that are linearly independent over the rational numbers $...
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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 ...
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Density of power of some transcendental number
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 ...
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