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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Prove that $\{x\}$ is transcendental base of $\mathbb{Q}(x,i)$

We have the field extension $\mathbb{C}/\mathbb{Q}$. Let $x\in \mathbb{R}\subset \mathbb{C}$ be transcendental over $\mathbb{Q}$. Show, that $\{x\}$ is a transcendental basis of $L = \mathbb{Q}(x,i)$. ...
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Question about purely transcendental extension

Let $F$ be a field of characteristic $\neq 2$ and let $u$ be transcendental over $F$. Suppose $u^2+v^2=1$. Show that $F(u,v)$ is a purely transcendental extension by showing that $F(u,v) = F(\frac{1+v}...
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Is there any $p \in \mathbb{Q}[x,y]$ with x and y degree both at least 1 such that $p(\pi,e)$ is known to be irrational?

The algebraic independence of $\pi$ and $e$ is a well known open problem, as is the specific case of rationality of $\pi + e$. My question is if there is any polynomial with rational coefficients (...
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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 ...
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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 ...
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100 views

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 $\sigma(\alpha)=\alpha$...
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A question about transcendental extensions

Consider an extension $K/F$. Suppose $S \subset K$ such that $S$ is algebraically independent over $F$. Prove that each $s \in S$ is transcendental over $F(S-\{s\})$. How do I prove this claim? Any ...
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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 $K$...
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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? [duplicate]

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?
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How can I prove that I can find $N$ real elements that is algebraically independent over $\mathbb{Q}$ for any N$?

I was wondering how can one show that there exist $N$ real elements that is algebraically independent over $\mathbb{Q}$ for any $N$? (I was thinking perhaps Lindemann–Weierstrass theorem can be used. ...
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Reference Request: Gelfond Schneider Theorem

I want to learn the background for the Gelfond Schneider Theorem: https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem I want to learn the background needed for the proof of this theorem, ...
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Field Extensions with Common Transcendental Numbers

Does anyone know if there is work done in this direction where one extends (the field) $\mathbb{Q}$ or $\bar{\mathbb{Q}}$ with certain common transcendental numbers such as $\pi$, $e$, etc. For ...
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Non-zero coefficient in a transcendence proof

I am studying the proof of the simple version of Lindemann's Theorem. Theorem. If $\alpha$ is a non-zero algebraic number, then $e^\alpha$ is transcendental. Apologies for the long read, but to ...
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How can I prove this question?

We know that it is not proved that $e^e$ is transcendental, so neither is the number that $e^{e\sqrt{2}}$. My question is, if one turns out to be, how can it be proved that the other is? Because there ...
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Is the sum of the inverses of the factorials of Fibonacci numbers transcendental?

This is the sum, $$e'=\sum_{n=1}^\infty \frac{1}{F_{n}!}$$ where $F_{n}$ is the $n^{th}$ Fibonacci number. Is it possible to prove that it will converge to a transcendental number? Edit: Proof of ...
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Transcendence of $\pi+\log\alpha$ and $e^{\alpha\pi+\beta}$

In the book "Transcendental Number Theory" by Alan Baker, he proves a few corollaries of Baker's theorem. I've attached this page below. After, he claims that special cases of these ...
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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 ...
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Explanation of proof of the transendence of $\alpha_1^{\beta_1} \dotsm \alpha_n^{\beta_n}$

I'm following this proof in the book "Transcendental Numbers" by M. Ram Murty and Purusottam Rath. The result is a corollary of Baker's theorem. There are a couple of things I don't ...
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What makes Hilbert's 7th problem important/relevant? [closed]

What was the motivation behind Hilbert's 7th problem? Looking into some of the history behind transcendental number theory, it seems that the field was almost non-existent in the late 1800's/early ...
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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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Proving equivalence between a summation expression and a power expression

I have that $$2^{-(a+1)!}(1+\frac{1}{2^{1}}+\frac{1}{2^{2}}+...)=2(2^{-a!})^{a+1}\,\,\,\,\,\,\,\,(1)$$ Which I am trying to show is $\geq \sum_{b=a+1}^{\infty}2^{-b!}$ in order to prove $\sum_{b=0}^{\...
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Prove $2^{\frac{1}{\log_2(x)}}$ is algebraic iff $x=2^n$ or $1/2^n$ for some $n \in \Bbb Q^+.$

Conjecture: $2^{\frac{1}{\log_2(x)}}$ is algebraic iff $x=2^n$ or $1/2^n$ for some $n\in \Bbb Q^+.$ How can I prove my conjecture? It might be very easy to prove but I am stuck at the moment. ...
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Do any ordered pairs of algebraic numbers satisfy this function?

Q: Does there exist an algebraic number $x,$ s.t. $f(x)$ is also an algebraic number? $f(x)=\exp\bigg(\frac{1}{\ln(x)}\bigg)$ for $x\ne0,1.$ I would like to prove that the set of points $(x,f(x))$ ...
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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 ...
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Prove that $e^{\frac{1}{\log(x)}}$ is at least countably transcendental

Q: Prove that $f(x)=e^{\frac{1}{\log(x)}}$ is at least countably transcendental for $x\in\Bbb R\cap (0,1).$
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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 $F$-...
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Clarification on the solution set of two surfaces being a hyperbola

$xy=e$ is a hyperbola. Looking at the LHS and RHS we have a hyperbolic parabaloid (a conoid) and a constant. Taken together, the equation yields a hyperbola. Manipulating algebraically, gives $xy=e^{\...
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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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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 $...
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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 ...
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What methods show that a number is transcendental?

I've been doing a lot of research on such theories lately and these are all I've found so far: Liouvilles criterion (here) Lindemann-Weierstrass theorem (here) Gelfond-Schneider theorem (here) ...
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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 ...
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Is the number transcendental?

Consider the following number: $R=\frac{1}{9}\sum^\infty_{n=1} 10^{-\frac{n\left(n+1\right)}{2}}\left(10^n-1\right)\left(n\left(\operatorname{mod}10\right)\right)$ =0....
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Rigidity of logarithms of positive integers

The following question is from my colleague. It seems to be emerged considering some elementary number theory problem. This is NOT as exercise or problem in published material although it may looks ...
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Flat $\mathbb{Z}$-modules and algebraic independence

I know that this question is naive, but I really want to find out if this observation is correct. First, let $\alpha_1, \alpha_2, \dotsc, \alpha_k$ be real numbers. Consider the $\mathbb{Z}$-module ...
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Density of power of some transcendental number

Let's note that where $\{x\}$ means fractional part of $x$. I am trying to figure out if $\{e^n\}_{n \in \mathbb{N}}$ and $\{\pi^n\}_{n \in \mathbb{N}}$ are dense in $[0,1]$. In general do we know if ...
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If the Liouville Constant is transcendental, is its exponentiation also transcendental?

So we have Liouville's Constant: $L_b=\displaystyle\sum_{n\in\mathbb{Z}^+}b^{-n!}=\left(0.\textbf{1}\textbf{1}000\textbf{1}00000000000000000\textbf{1}00000000...\right)_b$ And let $M$ be the ...
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Proving that the degree of transcendental extension is infinite

For a transcendental extension $K(\alpha):K$ for sub-field $K$ of $\mathbb{C}$, $[K(\alpha):K]=\infty$ Showing that the basis for $K(\alpha)$ that describes it as a vector space is infinite leads us ...
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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 ...
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Algebraic elements in a bijection between $p$-adic numbers and formal Laurent series over $\mathbb F_p$

Let $p$ be a prime number. We have a bijection $$\begin{align*}\sigma : \mathbb Q_p & \to \mathbb F_p(\!(t)\!) \\ \sum_{i \geq -n} a_i p^i & \mapsto \sum_{i \geq -n} a_i t^i \end{align*}$$ ...
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Does there exist a formula to calculate $2.357137939171\ldots$?

So I was messing with polynomials and I encountered the following equation: $$26214x^3 - 27761x^2 - 71019x - 21667 = 0.$$ Solving for $x$ using the cubic formula, I got three solutions (as expected, ...
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Density of $\{\sin(x^n)|n\in\mathbb{N}\}$ for $x>1$

While reading other topics, e,g, Is $n \sin n$ dense on the real line? or Is $\{ \sin n^m \mid n \in \mathbb{N} \}$ dense in $[-1,1]$ for every natural number $m$?, the following problem appeared in ...
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is the $\arcsin$ of a transcendental number, algebraic?

It's know that $\sin a$ is a transcendental $t$ if $a \neq 0$ and algebraic. So $\sin a = t$ This would imply $\arcsin(t) = a$ And this question can be done for all inverse trigonometry.
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What is know about the transcendence of inverse trigonometry and inverse hyperbolic functions?

I found on this wikipedia page https://en.wikipedia.org/wiki/Transcendental_number that $\sin a, \cos a, \tan a$ are transcendental numbers for $a \neq 0$ and algebraic. But there's no mention about ...
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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 ...
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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 ...
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Fields the closure of which is $\mathbb{C}$

Studying Galois Theory I have finally done the Fundamental Theorem of Algebra, which simply states that $\overline{\mathbb{R}} = \mathbb{C}$. My question is: do there exist other fields different ...
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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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48 views

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 ...