My general question is to find, if this is possible, two real numbers $a,b$ such that $K=\Bbb Q(a,b)$ is not a simple extension of $\Bbb Q$. $\newcommand{\Q}{\Bbb Q}$ Of course $a$ and $b$ can’t be both algebraic, otherwise $K$ would be a separable ($\Q$ has characteristic $0$) and finite extension, which has to be simple. […]

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?

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 independent over $\mathbb Q$? In other words, can we be sure that expressions like the following […]

Being motivated by this post, I was wondering if there is a proof (analogous to the case of Diophantine equations) that there is no general method for solving transcendental equations? It seems pretty clear, intuitively, that there can be no general method; but the only reason I feel strongly about that is because I can’t […]

What is known about irrationality measure of the Chaitin’s constant $\Omega$? Is it finite? Can it be a computable number? Can it be $2$?

Let $_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;c)$ denote the generalized hypergeometric function. Let $A \subset \mathbb R$ be the set of all values of $\ _pF_q(\cdot)$ at rational points $a_i,b_j,c\in \mathbb Q$. Since $A$ and its algebraic closure $\overline A$ are countable, there are obviously many constants $\alpha \notin \overline A$. Question: Are there any natural constants $\alpha$, as in […]

I can prove using the Gelfondâ€“Schneider theorem that the positive root of the equation $x^{x^x}=2$, $x=1.47668433…$ is an irrational number. Is it possible to prove it is transcendental?

Let $$x=0.112123123412345123456\dots $$ Since the decimal expansion of $x$ is non-terminating and non-repeating, clearly $x$ is an irrational number. Can it be shown whether $x$ is algebraic or transcendental over $\mathbb{Q}$ ? I think $x$ is transcendental over $\mathbb{Q}$. But I don’t know how to formally prove it. Could anyone give me some help ? […]

Can you prove that $e^{n\pi}$ is transcendental $\forall$ algebraic $n \in\mathbb{R}$ $n\neq $ 0 ? edit : n must be algebraic

In this question, I needed to assume in my answer that $e^{e^{e^{79}}}$ is not an integer. Is there some standard result in number theory that applies to situations like this? After several years, it appears this is an open problem. As a non-number theorist, I had assumed there would be known results that would answer […]

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