Questions tagged [transcendence-degree]

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If $G\subseteq GL_n(k)$ is an algebraic group of diagonal matrices, then $G$ is a torus isomorphic to a product of $\mathbb{G}_m$?

Part of proposition 3.1.9 of Geck's Algebraic Geometry and Algebraic Groups has the following setup. Let $G$ be a connected affine algebraic group over $k$ an algebraically closed field. If there ...
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Degree Calculation [on hold]

If we take from 1 to 9 as complete 360°(you can take as in number 360 too) rotation. from 1-4 how much degree rotation. kindly write clear calculation with well explained. Kind Regards
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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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1answer
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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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1answer
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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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1answer
632 views

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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1answer
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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 ...
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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 = \...
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1answer
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A field extension of degree $\leq 2$

Inspired by this question, I am asking the following question (with same notations): Let $K(ab,a+b) \subset K(a,b) \subset L$, where $a,b \in L$ and $K \subseteq L$ are fields of characteristic zero. ...
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1answer
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Let $\bar{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. What is the cardinality of a transcendental extension $K\setminus \bar{\mathbb{Q}}$?

Taking an algebraically closed field $K$ with $char(K)=0$ and $|K|\geq |\mathbb{C}|$, we have that $\bar{\mathbb{Q}}\subset K$. Let $T$ be a transcendence basis of $K\setminus \bar{\mathbb{Q}}$. ...
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Algebraicity of $R_1 \subseteq R_2$, where $R_1$ and $R_2$ have same finite Krull dimension

Assume that $R_i$ is a commutative $k$-algebra ($k$ is a field of characteristic zero) having finite Krull dimension $n_i$, $1 \leq i \leq 2$, and $k \subset R_1 \subseteq R_2$. Further assume that $...
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1answer
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Do How to convert degrees to decimal degrees?

Do How to convert degrees to decimal degrees? Example 1: 1. I have - 450 - degrees 2. We need get - 90 from 450 Example 2: 1. I have - 540 - degrees 2. We need get - 180 from 540
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1answer
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Transcendence degree of the fraction field of $k[G]$ for torsion free abelian group $G$

Let $k$ be a field of characteristic $p$ and $G$ be a torsion free abelian group . Then the group ring $k[G]$ is an integral domain , let $k(G)$ denote its field of fractions . Then can we say ...
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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^...
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1answer
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Two field extensions of transcendence degree 1

Let $K$ be a field, $K_1:=FRAC(K[x_1,\dots,x_n]/P)$ and $K_2:=FRAC(K[y_1,\dots,y_m]/Q)$ with prime ideals $P$ and $Q$, such that there is a field injection $\varphi:K_1\to K_2$. Assuming that the ...
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Reference for automorphisms of field extensions with non-zero transcendence degree

I'm looking for a book/reference that discusses automorphisms of field extensions which are not purely algebraic. Dummit and Foote only has a couple pages about this, and I wanted to learn more. I ...
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1answer
382 views

Understanding the dimension of a variety and transcendence degree

I am reading some introductory material on algebraic geometry and would like to understand the following statement: If a variety $V \subseteq \mathbb{A}^n$ is given by a single polynomial equation $...