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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.

135 questions
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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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What does algebraic independence mean?

If you search for algebraic independence, you will find the following on Wikipedia: In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation…
Kinheadpump
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Liouville-Roth Irrationality Measure of $\pi$ = 2 already proven?

I was looking through this paper and I was curious. Has it already been established that the Flint Hills series $\displaystyle\sum_{n\in\mathbb{Z}^{+}}\frac{\csc^2 n}{n^3}$ converges? And has it already been established that the Liouville-Roth…
user301661
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Transcendence measure for the canonical Liouville number

Let $\displaystyle\alpha=\sum_{n=0}^{+\infty}\frac1{10^{n!}}$. It is well-known that $\alpha$ is transcendental. I am looking for a transcendental measure for $\alpha$. That is exercise 11.15 of the book of Masser "Auxilliary polynomials in number…
joaopa
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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…
SSK
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