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.

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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 $(x,f(x))$ is the empty set, for algebraic $x.$ Are there any…
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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 $xy=e^{\big(\frac{1}{\ln(xy)}\big)}.$ So we have a…
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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 nowhere since it would deal with infinite…
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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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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 transcendental over K. If so, then in this question,…
B.A
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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 so that I can search for rigorous proof? Or…
Markoff Chainz
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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 $$\tr\deg(K_1K_2/F) \leqslant…
Marios Gretsas
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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 independent and $K$ is the maximal among all the…
Mary Star
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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 transcendental extension of $\mathbb{R}$ that is not…
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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 $K=\mathrm{Frac}(\mathbb{Z}[X_1,\ldots,X_n]/P)$, the field of fractions of the…
KDuck
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$V\subset \mathbb{P}^n(k)$ 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 $k(V)$ is the rational function field defined on $V$. The…
KittyL
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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.
Adnan
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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 sequence is transcendental. So from the definition,…
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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 may be some connection between the two...
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