Math 361, Spring 2014, Assignment 14

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

1. Splitting field (of a nonconstant polynomial $p\in F[x]$).
2. Galois group (of a nonconstant $p\in F[x]$).

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

1. Theorem on existence and uniqueness of splitting fields.
2. Theorem concerning finiteness of the Galois group.

Solve the following problems:

1. Give an example of a domain which is not a UFD.
2. Consider the polynomials $p=x^2-2, q=x^3-2,$ and $r=x^4-2$ in $\mathbb{Q}[x]$. Let $E_1, E_2,$ and $E_3$ be the splitting fields of $p, q,$ and $r$, respectively, over $\mathbb{Q}$. Compute the dimensions $[E_1:\mathbb{Q}], [E_2:\mathbb{Q}],$ and $[E_3:\mathbb{Q}]$. (Hint: follow the construction used to prove existence of splitting fields, and use the Dimension Formula. To compute the dimension of $E_3$, you may find the following fact helpful: $x^4-2 = (x^2 + \sqrt{2})(x^2-\sqrt{2}) = (x^2 + \sqrt{2})(x + \sqrt[4]{2})(x - \sqrt[4]{2})$.)
3. Compute the Galois group of $x^2+1\in\mathbb{R}[x]$.