Math 361, Spring 2014, Assignment 15

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

1. Fixed field (of a subgroup of a Galois group).
2. Pointwise stabilizer (of a subextension of a field extension).

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

1. Fundamental theorem of Galois Theory.
2. Galois' criterion for solvability by radicals.

Solve the following problems:

1. Compute $\mathrm{Gal}(\mathbb{C}/\mathbb{R})$. Then draw its subgroup diagram, the subextension diagram of $\mathbb{R}\rightarrow\mathbb{C}$, and arrows indicating the Galois correspondence.
2. Let $E=\mathbb{Q}[\sqrt[3]{2}]$. Compute $\mathrm{Gal}(E/\mathbb{Q})$. Then draw diagrams and arrows as in the previous problem. Why doesn't your answer contradict the Fundamental Theorem?
3. (Challenge problem) Taking $\mathbb{Q}$ as the base field, let $E$ be the splitting field of $p=x^3-2$. Compute the Galois group and draw diagrams illustrating the Galois correspondence. (Hint: Momentarily thinking of $E$ as a subfield of $\mathbb{C}$, one sees that complex conjugation swaps two of the roots of $p$. Moreover, it is a general theorem that if $p$ is irreducible over $F$ then $\mathrm{Gal}_F(p)$ acts transitively on the roots of $p$; this fact and the previous sentence are enough to determine the Galois group. For an added challenge, try to prove the transitivity theorem using the uniqueness property of the splitting field.)