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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Deduction from Schanuel's conjecture

I am deducing from the Schanuel's conjecture the following statement: Let $\alpha, \beta$ be positive real algebraic numbers with $\alpha,\beta \neq 1$ and $\frac{\log \beta}{\log \alpha} \notin \...
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Transcendence of meromorphic function vs formal power series

Consider the meromorphic function $f$ on $\mathscr D=\{z\in\mathbb C\mid|z|<1\}$ definied by $\displaystyle f(z)=\sum_{n\ge1}\frac{z^{2^n}}{z^{2^n}-\frac12}$. Obviously $f$ admits infinitely many ...
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House of the inverse

Let $\alpha,\beta\in\overline{\mathbb Q}$. Denote by $h(\alpha)$ the house $\alpha$, that is the maximum of $|\sigma(\alpha)|$ when $\sigma$ describes $\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)$. ...
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Transcendence of function and change of fields

Suppose the one has a sequence of rational functions $Q_n(z)\in\mathbb Q(z)$. Let $p$ be a prime number. Suppose that that there exists an infinite subset $X$ of $\mathbb Q_p$ such that: the ...
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What is $\text{trdeg}(F( X_i Y_j \mid 1 \leq i \leq n, 1 \leq j \leq m) / F)$?

We have indeterminate variables $X_i$ and $Y_j$ for $1 \leq i \leq n$ and $1 \leq j \leq m$. It is known $\text{trdeg}(L/F) = \text{trdeg}(L/K) + \text{trdeg}(K/F)$ for every field extension $F \...
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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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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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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}...
Hoang Nguyen's user avatar
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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 (...
Harrison Smith's user avatar
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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 ...
Derive Foiler's user avatar
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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 ...
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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?
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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 ...
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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 ...
Avi Mukesh's user avatar
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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 ...
Avi Mukesh's user avatar
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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 $(x,f(x))$ ...
John Zimmerman's user avatar
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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 $x\in\Bbb R\cap (0,1).$
John Zimmerman's user avatar
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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^{\...
John Zimmerman's user avatar
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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 $...
Mike Battaglia's user avatar
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3 answers
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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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