Math 361, Spring 2013, Assignment 7

The beginner...should not be discouraged if...he finds that he does not have the prerequisites for reading the prerequisites.

- P. Halmos

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

1. Meet (of two subgroups).
2. Join (of two subgroups).
3. Product (of two subgroups).

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

1. Classification of finite fields (Corollary 33.2 together with Theorems 33.10 and 33.12).
2. Theorem on irreducible polynomials over \(\mathbb{Z}_p\) (Corollary 33.11).
3. Theorem on the multiplicative group of a finite field (Theorem 33.5).
4. First isomorphism theorem (Theorem 34.2).
5. Second isomorphism theorem (Theorem 34.5).
6. Third isomorphism theorem (Theorem 34.7).

Solve the following problems:

• Section 33, problems 1, 3, and 8.
• Section 34, problems 1, 3, and 5.
• Construct a field with exactly eight elements (i.e. write down explicit addition and multiplication tables, and explain how you obtained these).