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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What algebraic function will result in transcendental function by indefinite integral?

Let $f(x)$ be a algebraic function over $\mathbb{C}$, under what condition will $\int f(x)dx$ be a transcendental function?
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How to prove that a logarithmic integral is transcendental?

After I've read, from [1], a proof that the logarithm $\log (x)$, defined for $x>0$, is a trancendental function, I wondered what should be the argument to prove (I believe that it holds) that the logarithmic integral is a trancendental function. I…
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How to compute an "effectively computable constant" in a formula of approximation of powers of $2$ and $3$

In his blog Terence Tao discusses the distance between powers of 2 and 3 and presents the following corollary: Corollary 4 (Separation between powers of {2} and powers of {3}) For any positive integers {p, q} one has $$ \displaystyle…
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Do the first differences of a b-normal number form a b-normal number? [Now with application!]

If it true that for every $b$-normal number (where the base $b,b^2,b^3,\ldots$ digits are asymptotically equiprobable) that the first differences of the base-$b$ digits, interpreted as a (signed) base-$b$ expansion, form a $b$-normal number? It…
Charles
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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 over $K(\tau)$, prove that: $1)$ $[K(\tau,…
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Five exponentials theorem

The six exponentials theorem is proved in most textbooks on transcendental number theory, and the four exponent conjecture is an open problem. Is there any good/accessible exposition of the five exponentials theorem?
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A step on the proof of Liouville's theorem on approximation

I'm having trouble following one step in the proof of Liouville's theorem on approximation of real algebraic numbers, from Murty and Rath's book "Transcendental Numbers". The step is: $$|\alpha-\frac{p}{q}|\geq…
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Integers (strictly) between 0 and 1 form the basis of transcendental number theory?

In a MathOverflow comment on the question of "What is the most useful non-existing object of your field?", an answer is given A number which is less than 1 and greater than 1. Which elicited a highly upvoted reply Integers (strictly) between 0…
Hooked
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Exceptional Set and Schanuel's conjecture

I was reading an article about transcendental funtions (Algebraic values of transcendental functions at algebraic points, by Huang, J., Marques, D., Mereb, M.). The authors gave an example that says: "Assuming Schanuel's conjecture to be true, it is…
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Irreducible polynomial in relatively algebraically closed extension

I am thinking about how to prove this fact: given a field extension $K\subseteq L$ such that $K$ is algebraically closed in $L$, and an irreducible polynomial $f\in K[X]$. Prove that $f$ is irreducible in $L[X]$. I tried by writing down possible…
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Are elements of the range of a transcendental function themselves transcendental, excepting a "few" special cases?

Let $f(x)$ be a transcendental function with $x\in\mathbb{C}$. Then are the values $f(x)$ themselves transcendental, except perhaps for a "few" exceptions? For example, it is known that $f(x)=e^x$ is transcendental, and clearly $f(0)=e^0=1$ is…
gone
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$t+1$ transcendent over $K$ if $t$ is transcendent

I've been solving some problems from my Galois Theory course and I want to check if the solution I came up with is correct. The question was: Given that an element $t$ is transcendent over a field $K$, is $t+1$ also transcendent over $K$? What I…
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How to prove that any rational linear combination of u and v will also be transcendent?

Let u a algebraic number and v a transcendent number, so any rational linear combination of u and v will also be transcendent. How to prove it?
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If $e^{ei\pi}$ turns out to be transcendent, can it be proved that $e^e$ is also?

We know that $e^e$ is not proved to be transcendent, so neither is the number $e^{ei\pi}$. If $e^{ei\pi}$ turns out to be transcendent, can it be proved that $e^e$ is also?
user892441
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Explanation of proof of transcendence of $e$

I'm following the proof for the transcendence of $e$ from the book "Transcendental Numbers" by M. Ram Murty and Purusottam Rath. I am struggling to understand the final few lines. As far as I understand, by the triangle inequality we…