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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Deducing Lindemann-Weierstrass from Baker's theorem

I'm aware that Baker's theorem with $n=1$ (for one algebraic number only) follows from that of Lindemann-Weierstrass. It is also often mentioned that Baker's result is a generalization of Lindemann-Weierstrass's. How do you prove that…
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Is this number a Liouville number?

Suppose I have a binary constant $q = 0.1010000000000000000000000000000000001001..._2$. In base 10 this number is $q $~$ .6250000000077325..$ and is defined as $$q = \sum_{\rho}^{\infty} \frac{1}{2^{\rho}}$$ Where $\rho$ is taken as the places…
Rob Bland
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Textbooks on transcendence theory

Is there a nice, modern textbook (some lecture notes or survey would do, too) that covers the main results and methods from transcendence theory? Ideally, it should also have some good exercises. So far I have been working mainly with the ($p$-adic)…
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Is it transcendental? Also normal?

The number we are considering is as follows: $0.a_1 a_2 a_3 \cdots $, where $a_{2n-1}=(n)_{(2)}, a_{2n}=(n)_{(3)}.$ So, the number is $$0.(1)(1)(10)(2)(11)(10)(100)(11)(101)(12)\cdots.$$ Is the number irrational? Is the number normal? Is the number…
hkju
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Representing complex numbers with nested exponentiation of rationals

Define $L_0=Q$ $L_1=\lbrace x \in C; e^{x} \in L_0 \rbrace$ $L_{-1}=\lbrace x \in C; \ln{x} \in L_0 \rbrace$ $L_{n+1}=\lbrace x \in C; e^{x} \in L_n \rbrace$ $0$ is in $L_1$ and $L_0$. Do any other numbers belong to more than one of these sets? Are…
Angela Pretorius
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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 must exploit maximality of $S$ somewhere. I took an…
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A relative description of algebraic and transcendental numbers

An algebraic number is a solution to a polynomial with rational coeficients over a field $K.$ A transcendental number is a number that is not algebraic. Has anyone proposed a relative description of algebraic and transcendental numbers? Working…
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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 $L = \mathbb{Q}(x,i)$. I've argued that $\mathbb{Q}(x)$ is a…
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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 $F(u,v) = F(\frac{1+v}{u})$. At first glance, my goal is to prove that…
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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 help will be appreciated.
user910011
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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. If we can find $N$ real linearly independent …
Johnny T.
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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, as well as the proof itself. But I don't know what…
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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 finite $n \in \mathbb{Z}$, does the following…
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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 $\sum_{b=0}^{\infty}2^{-b!}$ is transcendental by Liouville's…
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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. If for example $x=6$ then I think the expression…
geocalc33
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