Determining the smallest field in which a polynomial factors completely into linear terms. Solvability by Radicals:

Since $G$ is finite, we can average over $G$ to construct a $G$-invariant projection onto any $G$-invariant subspace of $V$. This shows that $\rho$ is completely reducible.

: A 13-page document containing selected solutions focused on automorphisms and field extensions.

Dummit And Foote Solutions Chapter 14 //top\\ Site

Determining the smallest field in which a polynomial factors completely into linear terms. Solvability by Radicals:

Since $G$ is finite, we can average over $G$ to construct a $G$-invariant projection onto any $G$-invariant subspace of $V$. This shows that $\rho$ is completely reducible. Dummit And Foote Solutions Chapter 14

: A 13-page document containing selected solutions focused on automorphisms and field extensions. Determining the smallest field in which a polynomial