Math 361, Spring 2017, Assignment 14

Carefully define the following terms, then give one example and one non-example of each:

1. Subnormal series.
2. Composition series.
3. Composition factors.
4. Solvable group.

Carefully state the following theorems (you do not need to prove them):

1. Galois' criterion for solvability by radicals.
2. Jordan-Hölder Theorem.
3. Theorem concerning simplicity of $A_n$.

Solve the following problems:

1. Section 35, problems 7 and 19.
2. Let $f$ be any non-constant polynomial in $\mathbb{Z}_p[x]$. Show that the equation $f=0$ is solvable by radicals. (Hint: the splitting field will be a finite-dimensional extension of $\mathbb{Z}_p$, hence a finite field. What is the Galois group?)

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