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.

Filter by
Sorted by
Tagged with
0
votes
0answers
38 views

Are separable transcendental extensions always separably generated?

The definition of separable I use is the following: a field extension $K/k$ is separable if for every extension $k'/k$, $k' \otimes_{k} K$ is a reduced $k$- algebra, and I know that this is equivalent ...
0
votes
2answers
43 views

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 ...
0
votes
1answer
33 views

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}^{\...
0
votes
1answer
37 views

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. ...
0
votes
0answers
27 views

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 $(x,f(x))$ ...
2
votes
0answers
39 views

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 ...
1
vote
1answer
37 views

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 $x\in\Bbb R\cap (0,1).$
1
vote
0answers
17 views

$F$-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 $F$-...
0
votes
0answers
31 views

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^{\...
2
votes
0answers
125 views

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 ...
0
votes
0answers
28 views

Can anyone suggest me a proof of Siegel-Shidlovsky Theorem?

I am searching for a proof of Sielgel-Shidlovsky Theorem about transcendence of E-functions. Can anyone tell me where can I find it? Or a proof of that the value of a E-function at algebraic number ...
2
votes
0answers
44 views

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 $...
2
votes
2answers
144 views

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 ...
7
votes
2answers
291 views

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) ...
3
votes
0answers
78 views

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 ...
7
votes
0answers
137 views

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....
3
votes
0answers
63 views

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 ...
3
votes
0answers
183 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 ...
1
vote
0answers
30 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 ...
3
votes
0answers
81 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 ...
0
votes
0answers
99 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 ...
3
votes
0answers
209 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 ...
5
votes
0answers
119 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 $$\begin{align*}\sigma : \mathbb Q_p & \to \mathbb F_p(\!(t)\!) \\ \sum_{i \geq -n} a_i p^i & \mapsto \sum_{i \geq -n} a_i t^i \end{align*}$$ ...
6
votes
1answer
141 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, ...
3
votes
0answers
95 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 ...
0
votes
3answers
145 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.
1
vote
1answer
82 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 ...
2
votes
0answers
99 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 ...
2
votes
1answer
103 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 ...
17
votes
2answers
218 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 ...
2
votes
1answer
58 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?
2
votes
0answers
43 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 ...
0
votes
0answers
488 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 $...
1
vote
1answer
246 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 ...
1
vote
1answer
114 views

About Schanuel's conjecture

If Schanuel's conjecture is true, why does it mean that $\pi$ and $e$ are algebricaly independent? I just understand that we have $deg.tr_{\mathbb{Q}} \mathbb{Q} (e,i\pi) \geq 2$.
1
vote
1answer
68 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 ...
7
votes
2answers
150 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 $x^x=e$? $x^x=π$?
1
vote
1answer
27 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 $...
-1
votes
2answers
228 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 ...
0
votes
1answer
73 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 ...
1
vote
1answer
64 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{\...
1
vote
2answers
146 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 ...
1
vote
1answer
137 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}\...
0
votes
1answer
45 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 ...
3
votes
1answer
73 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 ...
0
votes
0answers
198 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 $$...
0
votes
1answer
40 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 ...
1
vote
1answer
203 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
0
votes
1answer
331 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 ...
2
votes
1answer
78 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 ...