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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What is $\text{trdeg}(F( X_i Y_j \mid 1 \leq i \leq n, 1 \leq j \leq m) / F)$?
We have indeterminate variables $X_i$ and $Y_j$ for $1 \leq i \leq n$ and $1 \leq j \leq m$.
It is known $\text{trdeg}(L/F) = \text{trdeg}(L/K) + \text{trdeg}(K/F)$ for every field extension $F \...
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Let $L = F(S)$ for $S \subseteq L$. Does there exists a transcendental basis $T \subseteq S$ with $F(S) = F(T)$?
Let $L/F$ be a field extension with $L = F(S)$ for a subset $S \subseteq L$. My first question is:
Is there a subset $T \subseteq S$ such that $T$ is a transcendental basis?
Now let's say we have $I ...
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Degree of a field extension over $\mathbb{C}$ [closed]
I am supposed to find the degree of $ \mathbb {Q} (\sqrt3) \bigcap \mathbb {Q} (i)$ over $\mathbb{C}$.
My approach: Disclaimer: not sure how to go over $\mathbb{C}$.
The minimal poly of $ \mathbb {Q} ...
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Proving $[\mathbb{Q}(m(\alpha)):\mathbb{Q}]\leq[\mathbb{Q}(\alpha):\mathbb{Q}]$
Suppose $m$ is the minimal polynomial of a set $S$ of complex numbers such that for a complex number $\alpha\not\in S$, $\alpha$ is a root of the derivative of $m$. I want to prove that $[\mathbb{Q}(m(...
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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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Schanuel's Conjecture $\implies $ no surprises on the integers. $2^t+3^t=1 \implies t\notin\overline{\mathbb{Q}}$
Simple Question:
Is the proof given below correct?
Let $t$ satisfy $2^t+3^t=1$. We have
$t \approx -0.787884911025869783628555917298434738269083137354182194199 \dots$ according to wolfram alpha.
Then $...
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$L/K$ is algebraic iff its transcendence degree is zero
On Wikipedia, I read that
A field extension $L/K$ is algebraic if and only if its transcendence degree is $0$.
$[\Rightarrow]$ Suppose $L/K$ is an algebraic field extension, i.e., for every $l \in L$...
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Find a field extension $K=k(x,y)$ of transcendence degree 1 where $x\notin k(y)$, $y\notin k(x)$ and $K|k$ is purely transcendental
I saw this question online and it's actually a true or false question, the question is:
True of False: Let $K=k(x,y)$ be a field extension with transcendence degree 1. If $x\notin k(y)$ and $y\notin ...
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transcendental dimension of a variety
I am trying to understand the definition of the dimension of a variety using the notion of a transcendental basis. Consider for an algebraically closed field $\mathbb{K}$ the variety $V=\left\{(x,y)\...
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Add and multiplying disjoint algebraically independent sets are still disjoint algebraically independent sets
Let $T$ be a transcendence basis of $\Bbb R$ over $\Bbb Q$ and $F=\{A_n\subset T\colon n\in\Bbb N\}$ be a family of pairwise disjoint sets. Now, $$C=\{pA+q\colon A\in F \, and \,p,q\in Q\setminus\{0\}\...
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Extended of algebraically independent set
Let $T$ be a transcendental basis of $\Bbb R$ and $A\subset T.$ For $B\subset\Bbb R$, by $\overline{\Bbb Q}(B)$ the sets of all $x\in\Bbb R$ that are algebraic over $\Bbb Q(B).$
Now, let $S\subset\Bbb ...
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Is the subextension of a purely transcendental extension purely transcendental over the base field?
Let $K/E/F$ be extension of fields, where $K/F$ is purely transcendental.
It is generally not true that $K/E$ is purely transcendental. For example, take $F(x)/F(x^2)/F$. I wonder what is the ...
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Infinite transcendental basic
Given a field $K$ and infinite set $B$. Suppose that we have the transcendental degree of $K(B)$ over $K$ is $|B|$, i.e $[K(B):K]_t=|B|$. Can we conclude that $B$ is transcendental basic of $K(B)$ ...
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Algebraic independence of elementary symmetric polynomials
I am following the book Lectures on Algebra by Abhyankar, p. 638.
Let $k$ be a field, $x_1,x_2,\ldots,x_n$ be independent variables, and define
$$e_1=x_1+x_2+\cdots+x_n\\
e_2=\sum_{i<j} x_ix_j\\
\...
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Question about the proof that a function field over a $T_i$ field is a $T_{i+1}$ field
A field $K$ is a $T_i$ field if a system of $m$ polynomial equations with no constant terms in $n$ variables has a non-trivial solution over $K$ if the degrees $d_k, k \in \{ 1, \dots, m \}$ of the ...
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If $tr. deg. F/K=r$ then $tr.deg. F(x_1,\ldots,x_n)/K(x_1,\ldots,x_n)=r$.
Let $F/K$ be a field extension of transcendence degree $r$ and let $\{x_1,\ldots,x_n\}$ be a set of elements which are algebraically independent over $K$. Then is it true that $tr.deg. F(x_1,\ldots,...
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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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Krull Dimension of Affine Curve / Transcendence Degree of Coordinate Ring
I am trying to prove that the Krull dimension of an irreducible affine curve in $\mathbb{A}^2$ is 1. So my idea is a prove that the dimension of the coordinate ring is 1. So if the coordinate ring is
$...
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Transcendence basis as subset of generators
Let $K \subset L = K(a_1,...,a_n)$ be a field extension finitely generated as $K$-algebra with
transcendence degree $\operatorname{Trdeg}_K(L):= m \le n$. It is well
known that the choice of a ...
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$\operatorname{Trdeg}(\operatorname{Frac}(R/(a))) =\operatorname{Trdeg}(\operatorname{Frac}(R))-1$
Let $R$ be a commuative (noetherian?) integral domain and $Q=\operatorname{Frac}(R)$.
Assume that $\operatorname{Trdeg}(Q) := n < \infty$. Let $a \in R$
a nonzero prime element of $R$. I want to ...
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Proving that a variety has a lower dimension than its ambient space
I want to prove the following :
Let $V$ be an irreducible affine variety in $\mathbb{A}^n$ with $V \not=\mathbb{A}^n $. Then $dimV < n$.
I tried to prove this by contradiction but my proof doesn't ...
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How does transcendence degree relate to the closure of an open set?
Let $X$ be an irreducible quasiprojective variety. We define the dimension of $X$ to be the transcendence degree of $k(X)$ over $k.$
I'm trying to show that if $U \subset X$ is a nonempty open subset, ...
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About transcendence degree of an affine $K$-domain
Exercise 5.6 (G. Kemper)
If $A$ is an affine $K$-domain, then the transcendence degree of $A$ is the size
of a maximal algebraically independent subset of $A$.
Kemper's Proof:
Let $T\subseteq A$ be a ...
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Field of rational functions at a non singular point
Let $\mathbb{k}$ be an algebraically closed field, and let $p\subset \mathbb{k}[x_1,...,x_n]$ be a prime ideal contained in the ideal $(x_1,...,x_n)$. Suppose $0$ is a nonsingular point of the variety ...
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On a special kind of Discrete Valuation Ring [closed]
Let $(R,\mathfrak m, \kappa) $ be a Discrete Valuation Ring (i.e., a local PID) containing a field $k\hookrightarrow \kappa$ such that $\kappa$ has transcendence degree $1$ over $k$. Let $K$ be the ...
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Two algebraically closed fields are isomorphic if and only if they have the same transcendence degree over their prime fields.
Prove that two algebraically closed fields of the same characteristic are isomorphic if and only if they have the same transcendence degree over their prime
fields.
The prime field is isomorphism ...
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If $k\subset F\subset E $ is a tower of fields, prove that $\operatorname{tr\,deg}(E/k) =\operatorname{tr\,deg}(E/F) +\operatorname{tr\,deg}(F/k).$
If $k\subset F\subset E $ is a tower of fields, prove that $$\operatorname{tr\,deg}(E/k) = \operatorname{tr\,deg}(E/F) + \operatorname{tr\,deg}(F/k).$$
My attempt. Suppose that if $X$ is a ...
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Form of $u,v \in \mathbb{C}[x,y]$ satisfying $\mathbb{C}(s_1,s_2,k_1,k_2)=\mathbb{C}(u,v)$
Let $\beta$ be an involution on $\mathbb{C}[x,y]$, namely, $\beta$ is a $\mathbb{C}$-algebra automorphism of $\mathbb{C}[x,y]$ of order two.
Denote the set of symmetric elements with respect to $\beta$...
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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 ...
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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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2
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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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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 $...
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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[...
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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 ...
user525033