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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.

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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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0answers
111 views

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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20 views

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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71 views

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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13 views

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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184 views

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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81 views

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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1answer
125 views

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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54 views

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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3answers
98 views

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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1answer
46 views

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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77 views

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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1answer
93 views

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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2answers
211 views

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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1answer
48 views

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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37 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 ...
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290 views

How to handle the tensor product of fields of rational functions over $\mathbb{C}$?

Let $k$ be a field and let $s$ and $t$ be some transcendental indeterminates. This question was originally inspired by an exercise asking me to describe the product scheme of $\text{Spec }k(s)$ with $...
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1answer
207 views

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 ...
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72 views

About Schanuel's conjecture

If Schanuel's conjecture is true why it mean that $\pi $ ad $e$ are algebricaly independant ? I just understand that we have $deg.tr_{\mathbb{Q}} \mathbb{Q} (e,i\pi) \geq 2. $
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1answer
62 views

Is the set $\{X\sqrt{Y}, \sqrt{X}Y\}$ algebraically independent over $\mathbb{C}$?

Suppose $X$ and $Y$ are transcendental over some field $F$ (doesn't actually have to be $\mathbb{C}$, but I chose that for definiteness; I believe the answer will not depend on the exact field, as ...
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2answers
135 views

The solution of $x^x=2$ rational/algebraic irrational/transcendental?

What does the unique real number $x$ such that $x^x=2$ equal to? Is the value rational, algebraic irrational or transcendental? What about $x^x=3$? Or $x^x=e$? $x^x=π$?
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1answer
25 views

Can I found $x$ such that $K$ is a separable over $\textbf{F}_q(x)$?

Let $q$ be a power of a prime number, $K$ be a field containing $\textbf{F}_q$ and of transcendance degree one over it. Can I found $x\in K$ such that $K$ is a finite and separable field extension of $...
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2answers
171 views

Is my proof that $\gamma$ (the Euler-Mascheroni constant) is transcendental correct?

The Euler-Mascheroni constant $\gamma$ can be defined as $\lim\limits_{n\to \infty}(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}-\ln n)$. For every positive integer n (except for 1), the value of this ...
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1answer
55 views

Transcendental Extension decomposition

Say you have $K \subset L$ transcendental extension. I am wondering if the following is true: if $\exists M$ such that $K \subsetneq M \subsetneq L$ with L algebraic over M, then L is not purely ...
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1answer
59 views

Constructing a polynomial with rational coefficients which shares at least one root with a polynomial with algebraic coefficients in n variables.

my question can be seen as a extension to this question. Let $\overline{\mathbb{Q}}$ denote an algebraic closure of $\mathbb{Q}$. Given a polynomial with algebraic coefficients $f \in \overline{\...
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2answers
119 views

Algebraically dependent vs. one element can be expressed as a polynomial of the others

Let $k$ be a subfield of a field $X$, suppose $x_1, \cdots, x_n$ are algebraically dependent, that is there exists a non-zero polynomial with coefficients in $k$ and $p(x_1, \cdots x_n) = 0$. I ...
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1answer
125 views

Special cases of the Rohrlich-Lang conjecture with Gamma function

The Gamma function satisfies: $$\Gamma(z + 1) = z\Gamma(z)$$ $$\Gamma(z)\Gamma(1 - z) = \frac{\pi}{\sin(\pi z)}$$ $$\prod_{k = 0}^{n-1}\Gamma\left(a+\frac{k}{n}\right) = (2\pi)^{(n-1)/2}n^{-na + 1/2}\...
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1answer
40 views

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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1answer
64 views

$B$ 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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180 views

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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1answer
37 views

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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1answer
184 views

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
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1answer
224 views

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 ...
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1answer
72 views

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 ...
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1answer
32 views

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}.\...
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1answer
187 views

Excercise in Transcendental Number Theory

I am currently working through some of the content in Murty and Rath's Transcendental Numbers, and in their section entitled "Some Applications of Baker's Theorem" they present the following excercise:...
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3answers
330 views

Is $\Bbb Q(\sqrt 2, e)$ a simple extension of $\Bbb Q$?

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 ...
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1answer
75 views

What does Lindemann-Weierstrass-Theorem imply?

I try to understand the Theorem by Lindemann and Weierstrass: If $x,y$ are variables and I look at the field $\mathbf{Q}(x,y,exp(x),exp(y))$ what does L-W theorem imply on the transcendec degree of ...
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1answer
76 views

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 ...
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0answers
147 views

Deducing Lindemann-Weierstrass from Baker's theorem

I'm aware that Baker's theorem with $n=1$ (for one algebraic number only) follows from that of Lindemann-Weierstrass. It is also often mentioned that Baker's result is a generalization of Lindemann-...
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1answer
293 views

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-\...
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2answers
1k views

Is there an explicit irrational number which is not known to be either algebraic or transcendental?

There are many numbers which are not able to be classified as being rational, algebraic irrational, or transcendental. Is there an explicit number which is known to be irrational but not known to be ...
7
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1answer
121 views

$\mathrm{Aut}(\mathbb{Q}(\pi)/\mathbb{Q})=$?

Perhaps a silly question. I'm trying to understand trascendental field extensions, but I can't find a lot of instructive examples. Consider the extension $\mathbb{Q}(\pi)/\mathbb{Q}$. What is its ...
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1answer
116 views

Number of generators of ideal if quotient field has certain transcendence degree

I am trying to prove the following statement by induction on $n$: Let $P$ be a prime ideal of $\mathbb{Z}[X_1,\ldots,X_n]$ with $\mathbb{Z}\cap P = \{0\}$. Suppose that $K=\mathrm{Frac}(\mathbb{Z}[...
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2answers
827 views

Product of two transcendental numbers is transcendental

Let $\alpha,\beta$ be transcendental numbers. Which of the followings are true? 1)$\alpha\beta\ \text{ is transcendental}$. 2)$\mathbb{Q}(\alpha)\ \text{is isomorphic to }\mathbb{Q}(\beta)$ 3)$\...
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32 views

$V\subset \mathbb{P}^n(k)$ is irreducible, prove that $\dim(V)$ is the transcendence degree of $k(V)$ over $k$.

This is a problem from Ideals, Varieties, and Algorithms by Cox et. al. $V\subset \mathbb{P}^n(k)$ is irreducible, prove that $\dim(V)$ is the transcendence degree of $k(V)$ over $k$, where $k(V)$ ...
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2answers
293 views

For what values of $x$ is $\cos x$ transcendental?

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 ...
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1answer
262 views

Is this number a Liouville number?

Suppose I have a binary constant $q = 0.1010000000000000000000000000000000001001..._2$. In base 10 this number is $q $~$ .6250000000077325..$ and is defined as $$q = \sum_{\rho}^{\infty} \frac{1}{2^{\...
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0answers
44 views

Textbooks on transcendence theory

Is there a nice, modern textbook (some lecture notes or survey would do, too) that covers the main results and methods from transcendence theory? Ideally, it should also have some good exercises. So ...
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0answers
39 views

Continuity of Function involving logarithm function

I want to prove a function $f(x) = g(x) * log x $ is continuous on interval $[0, 1]$, where value of $g (x)$ is $0$ at lower limit point $0$. Anybody can help me out here. Thanks in advance.