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1) $G$ acts linearly on $W$ means that each element of $G$ acts by a linear transformation of $W$. Equivalently the action determines a homomorphism of $G$ into $GL(W)$. 2) This means a morphism of varieties, the algebraic group structure of $V$ isn't relevant here. 3) At the level of (closed) points, a morphism of varieties is a map that's locally given ...

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We can do this by two steps: The group $$\mathrm{SO}_{3}(\mathbb{C}) = \{A\in M_{3}(\mathbb{C}) \,|\, AA^{T} = I\}$$ naturally acts on the quadric $$Q = \{(x, y, z)\in \mathbb{C}^{3}\,|\, x^{2}+y^{2}+z^{2} =1\}$$ by $(x, y, z)^{T} \mapsto A(x, y, z)^{T}$. This action is transitive, and stabilier of $e_{1} = (1, 0, 0)^{T}$ is  \begin{pmatrix} 1 &...

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Since $K$ has characteristic $p$, it has no nontrivial $p$th roots of unity. The polynomial $x^p-1$ factors as $(x-1)^p$ so the only $p$th root of unity is $1$.

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An affine algebraic group $G$ is in particular an affine variety, therefore we know that $k[G]\cong k[X_1,\ldots,X_n]/\langle f_1,\ldots,f_m\rangle$ for some $n,m\in\mathbb{N}$ and some $f_1,\ldots,f_m\in k[X_1,\ldots,X_n]$. Set $I_G:=\langle f_1,\ldots,f_m\rangle$ and $x_i:=X_i+I_G$. Now, in light of the universal property of the ring of polynomials and in ...

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By using cofactor matrices, you can determine an expression of $i(A)$ as a rational function of the coefficient of $A$. Continuity of rational functions between affine algebraic sets

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I crossposted this question on MO, per Servaes' comment. It was answered by this user. Rosenlicht is using the Weil foundations for algebraic geometry, which were supplanted by Grothendieck's scheme theory in the 60's. A "point" in this context is a point in an affine patch, with coordinates in a universal domain, which is an algebraically closed extension ...

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