# 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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### Are separable transcendental extensions always separably generated?

The definition of separable I use is the following: a field extension $K/k$ is separable if for every extension $k'/k$, $k' \otimes_{k} K$ is a reduced $k$- algebra, and I know that this is equivalent ...
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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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### 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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### Can anyone suggest me a proof of Siegel-Shidlovsky Theorem?

I am searching for a proof of Sielgel-Shidlovsky Theorem about transcendence of E-functions. Can anyone tell me where can I find it? Or a proof of that the value of a E-function at algebraic number ...
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### From transcendental to algebraic

My question is given some set of transcendental numbers can we using algebraic operations form an algebraic number? My intuitive answer is no, could you please tell me what branch of mathematics it is ...
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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 ...
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### Transcendental basis of composite field

$\newcommand{\tr}{\operatorname{tr}}$Let $K_1,K_2$ field extension of the field $F$ which are contained in a larger field $E$. Prove that $\tr\deg(K_1K_2/F) \geqslant \tr\deg(K_i/F) ,i=1,2$ and ...
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### Is the union a basis of that extension?

Suppose that $H$ is a trancendental basis of the extension $A/F$ and $K$ is a trancendental basis of the extension $B/F$. So, $H$ is the maximal among all the subsets of $A$ that are $F$-algebraic ...
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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
### Transcendental extensions of $\mathbb{R}$ that are not purely transcendental
Purely transcendental extensions of $\mathbb{R}$ are those of the form $\mathbb{R}((X_i)_{i \in I})$ where $I$ is a set and the $X_i$'s are (distinct) indeterminates. Now, I wonder if there is a ...
### Transcendental elements in K with $\text{char}(K)=p>0$
Let $\Omega$ be an algebraically closed field with characteristic $p>0$, a subfield $K\subset \Omega$ and $L:=K(\tau^p, \eta^p)$. If $\tau$ is transcendental over $K$ and $\eta$ is transcendental ...