Math 361, Spring 2016, Assignment 13

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

1. Galois correspondence (i.e. the pair of maps $\gamma$ and $\phi$ that we discussed in class; as an example, compute these maps for some small field extension).
2. Normal extension.
3. Separable extension.
4. Galois extension.
5. Subnormal series (in a finite group).
6. Factors (of a subnormal series).
7. Solvable group.

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

1. Theorem relating the image of $\alpha\in E$ under an element of $\mathrm{Gal}(E/F)$ to the minimal polynomial $\mathrm{irr}(\alpha,F)$.
2. Theorem concerning separability of $F\rightarrow E$ when $F$ is finite or has characteristic zero.
3. Fundamental theorem of Galois theory.
4. Galois' criterion for solvability by radicals.

Solve the following problems:

1. Find at least two different subnormal series for the group $\mathbb{Z}_{12}$, and compute its factors. Is $\mathbb{Z}_{12}$ a solvable group?
2. Find a subnormal series for $S_2$ (the group of permutations on two letters), and compute its factors. Is $S_2$ a solvable group?
3. Repeat the previous problem with $S_3$ in place of $S_2$.
4. (Challenge) Repeat the previous problem for $S_4$.
5. (Hard challenge) Repeat the previous problem for $S_5$.

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