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 $K$.

# Questions tagged [transcendence-degree]

45 questions

**24**

votes

**1**answer

### 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 expressed as the ratio of two integers. This is…

Ozaner Hansha

- 701
- 4
- 15

**10**

votes

**2**answers

### Showing $[K(x):K(\frac{x^5}{1+x})]=5$?

Let $K$ be a field and $x$ be trnacendental over $K$. Compute
$[K(x):K(\frac{x^5}{1+x})]$.
I've never came across questions like these. It's easy to see that this degree is at most $5$, since: $$x^5-\alpha(x+1)=0$$ ($\alpha$ being…

35T41

- 3,246
- 8
- 24

**6**

votes

**2**answers

### Is the subextension of a purely transcendental extension purely transcendental over the base field?

Let $K/E/F$ be extension of fields, where $K/F$ is purely transcendental.
It is generally not true that $K/E$ is purely transcendental. For example, take $F(x)/F(x^2)/F$. I wonder what is the situation for $E/F$. Specifically, is $E/F$ purely…

fantasie

- 1,343
- 7
- 19

**6**

votes

**1**answer

### Algebraic independence of elementary symmetric polynomials

I am following the book Lectures on Algebra by Abhyankar, p. 638.
Let $k$ be a field, $x_1,x_2,\ldots,x_n$ be independent variables, and define
$$e_1=x_1+x_2+\cdots+x_n\\
e_2=\sum_{i

Maths Rahul

- 2,270
- 2
- 13

**6**

votes

**2**answers

### 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 it together.
If $P = P(t), Q = Q(t)$ are nonzero…

user525033

**4**

votes

**1**answer

### 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 all $k \in K$.
Now, I am confused:
(1) If $K…

user237522

- 6,071
- 3
- 10
- 21

**4**

votes

**0**answers

### Concerning $k \subset L \subset k(x,y)$

The following is a known result in algebraic geometry:
Let $k$ be an algebraically closed field of characteristic zero (for example, $k=\mathbb{C}$).
Let $L$ be a field such that $k \subset L \subset k(x,y)$
and $L$ is of transcendence degree two…

user237522

- 6,071
- 3
- 10
- 21

**4**

votes

**1**answer

### 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 dependent over $K$?
I.e: Are the monomials…

Pierre-Yves Gaillard

- 18,672
- 3
- 44
- 102

**3**

votes

**1**answer

### 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$$
\operatorname{trdeg}(A) = \sup\{|T| : T \subset A,\, T \text{ algebraically independent}\} \quad .
$$
Setting
$$
N = \sup\{|T|…

Carlos Esparza

- 3,573
- 10
- 32

**3**

votes

**1**answer

### Dimension of scheme of finite type over a field under base change (Hartshorne Ex. II.3.20)

Consider an integral scheme $X$ of finite type over a field $k$. If $k\subseteq k'$ is a field extension, then the scheme $X' = X\otimes_k k'$ is not necessarily integral. For instance, take $X = \operatorname{Spec} \mathbb{R}[x,y]/(x^2+y^2)$ over…

Davide Cesare Veniani

- 640
- 3
- 11

**2**

votes

**1**answer

### 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 $E$ over $k$ and $B=\{b_1,\ldots, b_m\}$ is a…

Sergey Guminov

- 2,732
- 1
- 9
- 16

**2**

votes

**0**answers

### 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 $\Bbb Q$, the field extension $\Bbb Q(z_1, ...…

Mike Battaglia

- 6,096
- 1
- 22
- 47

**2**

votes

**1**answer

### Degeneration of transcendence degree in polynomial rings

Before ask the general question, let us check the motivating example.
Consider two transcendental elements (and they are algebraically independent) over $\mathbb C$, say $x$ and $y$. Then $\mathbb C[x,y]$ has transcendence degree 2 over $\mathbb C$.…

LWW

- 694
- 3
- 12

**2**

votes

**0**answers

### 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 this part. Secondly, I'm asked to prove that the…

ReLonzo

- 147
- 7

**2**

votes

**0**answers

### 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 $\operatorname{Jac}(C,D)=0$.
(2) Assume that $F$ is of characteristic…

user237522

- 6,071
- 3
- 10
- 21