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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Extending the Lindemann Weierstrass Theorem

What I want to do is tinker with the Lindemann Weierstrass Theorem so I can ask 'what is so special about the number $e$'? I'll state the theorem below. Lindemann-Weierstrass Theorem (Baker's Reformulation) Let $\alpha_1, \dots, \alpha_n \in…
Mason
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$B$ spans algebraically $E$ over $F$

Let $E/F$ be an extension, $S=\{a_1,\ldots,a_n\}\subseteq E$ algebraically independent over $F$ and $S\subseteq T$, $T$ a subset of $E$, that spans $E$ algebraically over $F$. I want to show that there exists a set $B$ between $S$ and $T$, that is…
Mary Star
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For what values of $x$ is $\cos x$ transcendental?

For what values of $x$ is $\cos x$ transcendental? Is there any way I can figure out the values of $x$ where $\cos x$ is transcendental or do I have to check individually for every $x$ whether it is or not?
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Is there a general rule for proving that an equation has no analyticial solution

Somebody asked this here: Prove that an equation has no elementary solution But so far there is no response. The little math I know I have learnt it myself so I dont have a big picture of things. I was wondering if there is a general rule for…
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Borel's result on transcendence measure

In "Sur la nature arithmétique du nombre e" (Comptes rendus de l'Académie des Sciences 128 (1899), 596-9) Borel presented his result on transcendence measure for e. This can be restated as follows: For any positive $\varepsilon$ and for any…
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Maximum number of algebraically independent elements of a transcendental extension

As part of a course in Algebraic Geometry, I am trying to prove a corollary of a given lemma: Lemma: If $K$ is a field and $\alpha_1,\dots, \alpha_m$ are algebraically independent in $K$ and $\beta \in K$ is algebraically dependent of $\alpha_1,…
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transcendental number theory - classification

On Wikipedia one can find that transcendental number theory, or transcendednce theory is a branch of number theory. That confuses me a little since I thought that number theory is concerned with properties of positive integers and transcendence…
Adam
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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 immediate access to (like Roth's theorem), I…
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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 looking for books that are suitable for beginners. What…
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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 $\sigma(\alpha)=\alpha$, for all $\alpha \in F$. Here is my…
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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 containment is clear however I am not able to prove…
Akash Yadav
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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 $F$-automorphism of $F(B)$ ($B$ is a transcendence…
Ryze
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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 $\Bbb Q$, the field extension $\Bbb Q(z_1, ...…
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Preserving transcendence degree

Let $K$ be a field- either $\mathbb{C}$ or a finite field $\mathbb{F}_q$. Suppose $X_1,X_2,\dots,X_n$ are $n$ elements such that the extension field $$K(X_1,X_2,\dots,X_n)$$ has transcendence degree $k < n$ over $K$. In other words, there are some…
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Elementary result: If $m>n$, then any $f_1,...,f_m$ (non-zero polynomials) in $K[X_1,...,X_n]$ are algebraically dependent over $K$

I am looking into this proof: If $m$$>$ $n$, then any $f_1,...,f_m$ (non-zero polynomials) in $K[X_1,...,X_n]$ are algebraically dependent over $K$ ($K$ is a field). The proof starts by assuming that $f_1,...,f_m$ are algebraically independent over…