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.

119 questions
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The solution of $x^x=2$ rational/algebraic irrational/transcendental?

What does the unique real number $x$ such that $x^x=2$ equal to? Is the value rational, algebraic irrational or transcendental? What about $x^x=3$? Or $x^x=e$? $x^x=π$?
curious
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Excercise in Transcendental Number Theory

I am currently working through some of the content in Murty and Rath's Transcendental Numbers, and in their section entitled "Some Applications of Baker's Theorem" they present the following excercise: Suppose that the sum $$ F(z;x) =…
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$\mathrm{Aut}(\mathbb{Q}(\pi)/\mathbb{Q})=$?

Perhaps a silly question. I'm trying to understand trascendental field extensions, but I can't find a lot of instructive examples. Consider the extension $\mathbb{Q}(\pi)/\mathbb{Q}$. What is its group of automorphism,…
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Algebraic elements in a bijection between $p$-adic numbers and formal Laurent series over $\mathbb F_p$

Let $p$ be a prime number. We have a bijection $$\begin{align*}\sigma : \mathbb Q_p & \to \mathbb F_p(\!(t)\!) \\ \sum_{i \geq -n} a_i p^i & \mapsto \sum_{i \geq -n} a_i t^i \end{align*}$$ where the $a_i \in \{0, \ldots, p-1\}$. It maps pre-periodic…
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Does there exist a formula to calculate $2.357137939171\ldots$?

So I was messing with polynomials and I encountered the following equation: $$26214x^3 - 27761x^2 - 71019x - 21667 = 0.$$ Solving for $x$ using the cubic formula, I got three solutions (as expected, pursuant to the FTOA, namely, the Fundamental…
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Density of $\{\sin(x^n)|n\in\mathbb{N}\}$ for $x>1$

While reading other topics, e,g, Is $n \sin n$ dense on the real line? or Is $\{ \sin n^m \mid n \in \mathbb{N} \}$ dense in $[-1,1]$ for every natural number $m$?, the following problem appeared in my head: is $\{\sin(x^n)|n\in\mathbb{N}\}$ dense…
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square-root of a transcendental number

I know that a square-root of an irrational number is also irrational. Is it also true that the square root of a transcendental number is transcendental?
Adam
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A notion of transcendence degree for fields of formal power series?

There is an intuitive sense in which one would like to say that the field of formal Laurent series $k((z))$ over a field $k$ has "transcendence degree $1$". Of course, it doesn't have transcendence degree $1$ over $k$ in the usual sense (in fact I…
Bruno Joyal
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Product of two transcendental numbers is transcendental

Let $\alpha,\beta$ be transcendental numbers. Which of the followings are true? 1)$\alpha\beta\ \text{ is transcendental}$. 2)$\mathbb{Q}(\alpha)\ \text{is isomorphic to }\mathbb{Q}(\beta)$ 3)$\alpha^\beta\ \text{is transcendental }$ 4)$\alpha^2\…
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When does the following construction generate a transcendental number?

Given $n\in[0,1]$ with base-b expansion $0.n_1n_2n_3\dots$, define $\Delta_b(n)$ to be the number with the following base-b expansion: $\huge{ 0.\underbrace{n_1}_{1^{st}\text{ block}}\overbrace{n_1n_2}^{2^{nd}\text{…
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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 question cannot be generalised to any integer $k$,…
Adam Rubinson
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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 corollaries give the transcendence of $\pi+\log\alpha$ for…
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Understanding the Fundamental Theorem of Symmetric Polynomials within the context of proving $\pi$ transcendental

I am currently studying the proof of the transcendence of $\pi$. There are a bunch of proofs scattered across the web (here, here, and here, to list some); some derive from the Lindemann-Weierstrass Theorem, while some are given standalone. But all…
Coffee_Table
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Are logarithms of prime numbers algebraically independent?

From Baker's theorem it follows that a linear combination of natural logarithms of prime numbers with non-zero algebraic coefficients can never be zero. Has it been proved that the set of all natural logarithms of prime numbers is algebraically…
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Prove that $E_0$ is transcendental

Consider the non-negative natural numbers: $0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19\dots$ Encode the primes as $1$, the rest as $0$. $E = 0,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1\dots$ The number $E_0$ is $001.10101000101000101\dots$ (i.e.…