Math 361, Spring 2013, Assignment 14

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

1. Splitting field of a polynomial.
2. Solvability by radicals.
3. Galois group of a field extension.
4. Galois correspondence (i.e. the mappings that transform subextensions of a field extension to subgroups of its Galois group, and vice versa).

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

1. Theorem on the existence of splitting fields.
2. Theorem on uniqueness of splitting fields.
3. Galois' criterion for solvability by radicals.

Solve the following problems:

1. Let \(\mathbb{Q}\rightarrow E\) be the splitting field of \(x^3-2\). Compute the dimension \([E:\mathbb{Q}]\). Compare this with the order of the Galois group.
2. Compare the dimension of \(\mathbb{R}\rightarrow\mathbb{C}\) to the order of its Galois group. Can you make a general conjecture about how the dimension of a field extension is related to the order of its Galois group?
3. Compare the dimension of \(\mathbb{Q}\rightarrow\mathbb{Q}(\sqrt[3]{2})\) to the order of its Galois group. Do you wish to modify the conjecture you made above? (The conjecture suggested by the two example above is true for all "nice" field extensions, specifically for those that are finite-dimensional, normal, and separable. The definitions of these terms can be found in any book on Galois theory.)
4. (Challenge problem) As above, let \(\mathbb{Q}\rightarrow E\) be the splitting field of \(p=x^3-2\). To fix notation in what follows, let the roots of \(p\) in \(E\) be \(\alpha_1, \alpha_2\) and \(\alpha_3\). Recall that the Galois group of \(E/\mathbb{Q}\) is isomorphic to the group of permutations of the set \(\{\alpha_1,\alpha_2,\alpha_3\}\).
   1. List all subgroups of the Galois group.
   2. For each such subgroup \(H\), find a basis for the fixed field \(\phi(H)\).
   3. Compare the dimension of each fixed field to the index of the subgroup that gave rise to it. Can you make any general conjecture (and least for "nice" field extensions)?