# Questions tagged [transcendence-degree]

In abstract algebra, the transcendence degree of a field extension $L/K$ is a certain rather coarse measure of the “size” of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of $L$ over $K$.

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### finite extension of $\mathbb{F}_p(x)$ must of the form $\mathbb{F}_{p^r}(t)$?

Let $p>0$ be a prime. Let $K$ be a finite extension of $\mathbb{F}_p(x)$. Do we must have $K\cong \mathbb{F}_{p^r}(t)$ for some $r\geq 1$?
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### Properties of field extensions of finite transcendence degree

Let $K/F$ be a field extension of transcendence degree $n\ge 2$. If $E$ is a subfield of $K$ such that $F\subset E \subset K$ and the transcendence degree of $E$ over $F$ equals $n-1$, then $K$ is a ...
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### Transcendence degree of $F(x)$ over a field $F$

I'm studying Algebraic geometry and have been stuck at processing the concept of Transcendental degree of a polynomial and came across the following argument online. The transcendence degree of the ...
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### How is the field extension finite

I was reading the book Arithmetic of Elliptic Curves by Silverman and I came across the below proposition Proposition.Let C/K be a curve(Projective variety of dimension 1), and let t ∈ K(C) be a ...
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### Is $\mathbb{R}(x,\sqrt{1+x^2})$ a purely transcendental extension of $\mathbb{R}$?

I'm working on Exercise 9, Section 19 of Morandi's Field and Galois Theory: If $K=\mathbb{R}(x,\sqrt{1+x^2})$, show that there is a $t\in K$ with $K=\mathbb{R}(t)$. Trying to deduce a workable $t$, ...
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### If $K=\mathbb{C}(x)(\sqrt{-1-x^2})$, then $K=\mathbb{C}(t)$ where $t=(i-x)^{-1}\sqrt{(-1-x^2)}/(i-x)$?

This is exercise 7 of Section 19 on Transcendence Degrees in Morandi's Field and Galois Theory. Let $K=\mathbb{C}(x)(\sqrt{-1-x^2})$. Show that $[K:\mathbb{C}(x)]=2$, and show that $K=\mathbb{C}(t)$ ...
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### $(r+1)^{\frac{1}{3}}$ transcendental and $s^2-s$ algebraic on $\mathbb{Q}$ then $r$ is transcendental and $s$ algebraic on $\mathbb{Q}$

Let $r \in \mathbb{R}^+$. $(1)$ Show that $r$ is transcendental on $\mathbb{Q} \iff (r+1)^{\frac{1}{3}}$ is transcendental on $\mathbb{Q}$, and that $(2)$ $r$ is algebraic on $\mathbb{Q} \iff r^2-r$ ...
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### Proper subfields of $\mathbb{C}$ isomorphic to $\mathbb{C}$

It is known that $\mathbb{C}$ has proper subfields which are isomorphic to $\mathbb{C}$, see this question; let $K$ be such subfield of $\mathbb{C}$. Let $\iota: K \to \mathbb{C}$, $\iota(k)=k$ for ...
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### Minimal polynomial of $b$ over $k[a+b,c+d]$

Let $k$ be a field of characteristic zero. Let $a,b,c,d \in k[x,y]$ be four polynomials, each two are algebraically independent over $k$. Let $R:=k[a+b,c+d]$ be of transcendence degree two over $k$ (...
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### Transcendence degree and Krull dimension of finitely generated algebras

Let $K$ be a field, and let $a_1,\dots,a_{n+1}$ be $n+1$ elements of a finitely generated $K$-algebra $A$ of Krull dimension $n$. Are the elements $a_1,\dots,a_{n+1}$ always algebraically ...
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### Proof verification: transcendence degree additive in towers

I am trying to prove that if $k\subseteq E\subseteq F$ are field extensions, then $$\text{tr.deg}_k F=\text{tr.deg}_k E+\text{tr.deg}_E F.$$ If $A=\{a_1,\ldots, a_n\}$ is a transcendence basis for $E$...
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### Dimension of product of varieties

I've got this exercise asking me to prove first that the product of quasi-projective varieties $X$ and $Y$ (henceforth just "varieties") is irreducible iff both $X$ and $Y$ are. I managed to solve ...
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### Is every affine variety birational to affine space?

Is it true that any affine variety is birational to affine space of the proper dimension? For example, say I have an affine curve in $\mathbb{A}^n$, it makes sense that this curve should be birational ...
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### transcendence basis of field extensions of $\mathbb{Q}$

In some exercice, I see the following: Let $K = \mathbb{Q}(X_1 ,\dots , X_n )$ and $k = \mathbb{Q}(e_1 , \dots, e_n )$, where $(e_i)$ are the elementary symmetric polynomials. It states: "Since K ...
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### For $C,D \in \mathbb{Z}[x,y]$: $\operatorname{Jac}(C,D)=0$ if and only if $C$ and $D$ are algebraically dependent over $\mathbb{Z}$?

The following is a known result due to Carl Gustav Jacob Jacobi (1841): Let $F$ be any field, $C,D \in F[x,y]$. (1) If $C$ and $D$ are algebraically dependent over $F$, then $\operatorname{Jac}(C,D)=0$...
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### Can you determine the transcendence degree of an algebra by looking at a generating set?

Let $K$ be a field and $A$ be a $K$-algebra generated (as $K$-algebra) by a set $S$. The transcendence degree of $A$ is \operatorname{trdeg}(A) = \sup\{|T| : T \subset A,\, T \text{ algebraically ...
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### If $E$ is algebraic extension field of $F$, then also $E(x)$ is algebraic extension of $F(x)$?

I had a so simple question. Question: Let $E$ be an algebraic extension field of a field $F$. Does it follow that $E(x)$ is an algebraic extension field of $F(x)$? I think it is not true and i ...
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### Are all transcendental numbers a zero of a power series?

So I came across the concept of extending the notion of irrationality to higher degree polynomials. The base case of this is standard irrationality. That is, a number is irrational if it cannot be ...
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### Non-injective polynomial map $\mathbb{R}^2 \to \mathbb{R}^2$, injective on lines

Let $F: \mathbb{R}^2 \to \mathbb{R}^2$, $(x,y) \mapsto (f(x,y),g(x,y))$, where $f(x,y),g(x,y) \in \mathbb{R}[x,y]$, each is of $(1,1)$-degree at least one, and $f(x,y),g(x,y)$ are algebraically ...
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### Lüroth's theorem for transcendence degree two

Let $k$ be an algebraically closed field of characteristic zero, and $k \subsetneq L \subseteq k(x_1,\ldots,x_n)$ a field of transcendence degree two over $k$. According to the comments of ulrich and ...
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### Show that $[k(t): k(t^4 + t) ] = 4$

Let $k$ be the field with $4$ elements, $t$ a transcendental over $k$, $F = k(t^4 + t)$ and $K = k(t).$ Show that $[K : F] = 4.$ I think I have to use the following theorem, but I'm not quite putting ...