Questions tagged [transcendence-degree]

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22 questions
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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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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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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 ...
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 \$...