Math 361, Spring 2015, Assignment 10

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

1. Finite field.
2. Splitting field (of a polynomial $f\in F[x]$).

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

1. Theorem concerning the possibility of squaring the circle, duplicating the cube, or trisecting general angles with compass and straightedge.
2. Theorem concerning the existence and uniqueness of splitting fields.

Solve the following problems:

1. Section 32, problem 3.
2. Section 33, problems 1 and 3.
3. Construct the splitting field of $x^3 - 3$ over $\mathbb{Q}$. In particular, compute its dimension over $\mathbb{Q}$.
4. Construct the splitting field of $x^3 - 1$ over $\mathbb{Q}$. In particular, compute its dimension.
5. The splitting field of $x^n - 1$ over $\mathbb{Q}$ is called the $n$th cyclotomic field. Investigate the structure of this field for several more small values of $n$.

--------------------End of assignment--------------------