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.

Primarily, questions related to determination of transcendental numbers, theory of algebraic independence and diophantine approximation should have this tag. Closed form-related question can have this tag if is broad enough. Example: is $\text{K} \cap \mathbb{Q} = \lbrace\emptyset\rbrace$ for the ring $\text{K} \in \mathbb{C}$ defined as $(...)$?

Also known as transcendental number theory.

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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 understand. Why is it sufficient to show that…
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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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Operations with cardinals in transcendence base proof

I want to prove the theorem that two transcendence bases for a transcendental field extension have the same cardinality. I've come with this situation : if $A$ and $B$ are two transcendence bases for $E/F$ a field extension and $A$ is infinite, for…
Patrick Da Silva
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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 fractional part of a power of a transcendental…
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What is know about the transcendence of inverse trigonometry and inverse hyperbolic functions?

I found on this wikipedia page https://en.wikipedia.org/wiki/Transcendental_number that $\sin a, \cos a, \tan a$ are transcendental numbers for $a \neq 0$ and algebraic. But there's no mention about their inverse, $\arcsin, \arccos, \arctan$. I will…
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How to handle the tensor product of fields of rational functions over $\mathbb{C}$?

Let $k$ be a field and let $s$ and $t$ be some transcendental indeterminates. This question was originally inspired by an exercise asking me to describe the product scheme of $\text{Spec }k(s)$ with $\text{Spec }k(t)$. This obviously comes down to…
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About Schanuel's conjecture

If Schanuel's conjecture is true, why does it mean that $\pi$ and $e$ are algebricaly independent? I just understand that we have $deg.tr_{\mathbb{Q}} \mathbb{Q} (e,i\pi) \geq 2$.
Pierre21
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Is the set $\{X\sqrt{Y}, \sqrt{X}Y\}$ algebraically independent over $\mathbb{C}$?

Suppose $X$ and $Y$ are transcendental over some field $F$ (doesn't actually have to be $\mathbb{C}$, but I chose that for definiteness; I believe the answer will not depend on the exact field, as long as both $X$ and $Y$ are transcendental over…
Tob Ernack
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Can I found $x$ such that $K$ is a separable over $\textbf{F}_q(x)$?

Let $q$ be a power of a prime number, $K$ be a field containing $\textbf{F}_q$ and of transcendance degree one over it. Can I found $x\in K$ such that $K$ is a finite and separable field extension of $\textbf{F}_q(x)$? Many thanks!
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Constructing a polynomial with rational coefficients which shares at least one root with a polynomial with algebraic coefficients in n variables.

my question can be seen as a extension to this question. Let $\overline{\mathbb{Q}}$ denote an algebraic closure of $\mathbb{Q}$. Given a polynomial with algebraic coefficients $f \in \overline{\mathbb{Q}}[X_1,...,X_n]$, say…
cdwe
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Algebraically dependent vs. one element can be expressed as a polynomial of the others

Let $k$ be a subfield of a field $X$, suppose $x_1, \cdots, x_n$ are algebraically dependent, that is there exists a non-zero polynomial with coefficients in $k$ and $p(x_1, \cdots x_n) = 0$. I guess it does not imply that for some $x_i$ there…
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Special cases of the Rohrlich-Lang conjecture with Gamma function

The Gamma function satisfies: $$\Gamma(z + 1) = z\Gamma(z)$$ $$\Gamma(z)\Gamma(1 - z) = \frac{\pi}{\sin(\pi z)}$$ $$\prod_{k = 0}^{n-1}\Gamma\left(a+\frac{k}{n}\right) = (2\pi)^{(n-1)/2}n^{-na + 1/2}\Gamma(na)$$ The Rohrlich conjecture states: Any…
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Is $e^{n\pi}$ transcendental?

Can you prove that $e^{n\pi}$ is transcendental $\forall$ algebraic $n \in\mathbb{R}$ $n\neq $ 0 ? edit : n must be algebraic
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Algebraic Independence of Functions in Several variables

If we have $n$ algebraic numbers $x_1,x_2,...,x_n$ $\in$ $\bar{\mathbb{Q}}^d$ which are linearly independent over $\mathbb{Q}$. How do we show that the $n$ functions $f_i(z_1,z_2,...,z_d)= e^{{x_i}.\bar{z}} $ , $1 \leq n$ are algebraically…
Suraj
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What does Lindemann-Weierstrass-Theorem imply?

I try to understand the Theorem by Lindemann and Weierstrass: If $x,y$ are variables and I look at the field $\mathbf{Q}(x,y,exp(x),exp(y))$ what does L-W theorem imply on the transcendec degree of this field? Is it 4?