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Sean Lawton
Sean Lawton
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InReference: http://www.math.u-psud.fr/~colliot/mumbai04.pdf

Proposition 4.3. on page 18 in the above reference reads as follows: Assume $k = \overline{k}$. If $V$ is a finite dimensional vector space over $k$ and $G \subset GL(V)$ is an (abstract) abelian group consisting of semisimple elements elements, then $k(V)^G$ is pure.

I would like to find an abelian group $G \subset GL(V)$ such that $k(V)^G$ is not pure (if it exists it would need to be infinite due to FischerFischer's theorem, and nonnot a connected solvable group according to propositionProposition 4.4).

Thanks.

In http://www.math.u-psud.fr/~colliot/mumbai04.pdf

Proposition 4.3. page 18 reads as follows: Assume $k = \overline{k}$. If $V$ is a finite dimensional vector space over $k$ and $G \subset GL(V)$ is an (abstract) abelian group consisting of semisimple elements, then $k(V)^G$ is pure.

I would like to find an abelian group $G \subset GL(V)$ such that $k(V)^G$ is not pure (if it exists it would need to be infinite due to Fischer theorem, and non connected solvable group according to proposition 4.4).

Thanks.

Reference: http://www.math.u-psud.fr/~colliot/mumbai04.pdf

Proposition 4.3. on page 18 in the above reference reads as follows: Assume $k = \overline{k}$. If $V$ is a finite dimensional vector space over $k$ and $G \subset GL(V)$ is an (abstract) abelian group consisting of semisimple elements, then $k(V)^G$ is pure.

I would like to find an abelian group $G \subset GL(V)$ such that $k(V)^G$ is not pure (if it exists it would need to be infinite due to Fischer's theorem, and not a connected solvable group according to Proposition 4.4).

Thanks.

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Portland
Portland
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Is the field of invariants $k(V)^G$ purely transcendental over $k$?

In http://www.math.u-psud.fr/~colliot/mumbai04.pdf

Proposition 4.3. page 18 reads as follows: Assume $k = \overline{k}$. If $V$ is a finite dimensional vector space over $k$ and $G \subset GL(V)$ is an (abstract) abelian group consisting of semisimple elements, then $k(V)^G$ is pure.

I would like to find an abelian group $G \subset GL(V)$ such that $k(V)^G$ is not pure (if it exists it would need to be infinite due to Fischer theorem, and non connected solvable group according to proposition 4.4).

Thanks.