0
votes
1answer
28 views

how fast does the proportion of associative operations on $S$ decrease with |$S$|?

as doubtless many have done before me, i recently fell into wondering how many of the binary operations on a finite set are associative. the stackexchange software fortunately pointed me to this ...
0
votes
2answers
48 views

Element in field of quotients is transcendental

Let $F\subseteq E$ be fields, and let $c\in E$. Let $F(c)$ be the field of quotients containing $F$ and $c$. Suppose $c$ is transcendental over $F$. Prove that every element in $F(c)$ but not in ...
2
votes
2answers
110 views

Uncountable sets of transcendental numbers

As a sort of follow up question to a previous question found here, besides the Liouville numbers, are there any other uncountable collections of transcendental numbers that are known? Clearly you ...
1
vote
1answer
39 views

Question about the proof that $K(u)$ is isomorphic to $K(x)$ if u is transcendental over $K$.

I have a few questions about a proof of the statement that if $u$ is transcendental over $K$ then $K(u)\cong K(x)$. My questions are marked as red and stated below the proof. The proof goes like ...
2
votes
2answers
94 views

Extend a rational number field $\mathbb{Q}$ by using a transcendental number?

Here denoting a set of real transcendental numbers $\mathbb{T}$, what can we then say about the structure $$ \mathbb{Q}(t) = \left\{\, \sum_{k=0}^{+ \infty} a_k t^k\mathrel{}\middle|\mathrel{} a_k \in ...
1
vote
0answers
59 views

Can you make some sort of structure out of transcendental numbers?

Let $T$ be the set of trancendental numbers over $\Bbb{Q}$. Then it is an easy proof that for all $a,b \in T$, either $a + b$ or $ab$ or both are transcendental. What if you defined the operation ...
1
vote
1answer
186 views

Is $a^b$ transcendental when $a$ and $b$ are?

I am being asked this question as an exercise in Garling's "A course in Galois theory". But isn't this an open question in math?
38
votes
11answers
2k views

What if $\pi$ was an algebraic number? (significance of algebraic numbers)

To be honest, I never really understood the importance of algebraic numbers. If we lived in an universe where $\pi$ was algebraic, would there be a palpable difference between that universe and ours? ...
0
votes
1answer
62 views

number of the form $\frac{a_0+a_1\pi +\text{…}+a_n\pi ^n}{b_0+b_1\pi +\text{…}+b_m\pi ^m}$ [closed]

all numbers of the form $$\begin{align*}\frac{a_0+a_1\pi +\text{...}+a_n\pi ^n}{b_0+b_1\pi +\text{...}+b_m\pi ^m}\end{align*}$$ form an number field. $n,m$ are arbitrary non-negative itegers, ...
4
votes
2answers
173 views

How to show $e^{2 \pi i \theta}$ is not algebraic.

I was wondering if someone could possibly help me figure out how to show $e^{2 \pi i \theta}$ is not algebraic when $\theta$ is irrational. Thanks!
2
votes
2answers
251 views

The definition of “algebraically independent”

In Lang's Algebra, he gives a definition that Elements $x_1, \cdots, x_n\in B$ are called algebraically independent over $A$[a subring of $B$] if the evaluation map $$f\mapsto f(x)$$is injective. ...
6
votes
2answers
552 views

Can $\pi$ be a root of a polynomial under special cases?

What if we consider polynomials whose coefficients are either rational or $e$, that is, a polynomial in $\mathbb{Q} \cup \{e\}$ with $\pi$ as a root. Can this happen? Does it matter if we change ...